FE Exam Flashcards: Engineering Equations, Units & Handbook Skills

Cartões deste baralho

1. Cartão 1

   Pergunta

   SI base unit of length?

   Resposta

   The metre, symbol $m$.

2. Cartão 2

   Pergunta

   Slope between two points?

   Resposta

   $$m=\frac{y_2-y_1}{x_2-x_1}$$

   This requires $x_2\ne x_1$; slope is the change in the dependent variable per unit change in the independent variable.

3. Cartão 3

   Pergunta

   Net force relationship for constant mass in an inertial frame?

   Resposta

   $$\sum\mathbf{F}=m\mathbf{a}$$

   Use a coherent unit system and include only external forces on the chosen body.

4. Cartão 4

   Pergunta

   Cash-flow sign convention from the decision maker's viewpoint?

   Resposta

   Receipts are positive and disbursements are negative. Choose one viewpoint and keep it consistent across the cash-flow diagram and equations.

5. Cartão 5

   Pergunta

   Where should an FE candidate get the current reference handbook for practice?

   Resposta

   From the candidate's MyNCEES account. Use the current supplied file rather than an old indexed copy or a third-party mirror.

6. Cartão 6

   Pergunta

   Probability of the complement of event $A$?

   Resposta

   $$P(A^c)=1-P(A)$$

7. Cartão 7

   Pergunta

   An engineering judgment conflicts with pressure to protect schedule or cost. First duty?

   Resposta

   Protect public health, safety, and welfare. Document the technical concern and escalate through appropriate channels.

8. Cartão 8

   Pergunta

   SI base unit of mass?

   Resposta

   The kilogram, symbol $kg$. It is a mass unit, not a force unit.

9. Cartão 9

   Pergunta

   Linear equation from slope and intercept?

   Resposta

   $$y=mx+b$$

   Here $m$ is slope and $b$ is the value of $y$ where $x=0$.

10. Cartão 10

    Pergunta

    Velocity after time $t$ under constant acceleration?

    Resposta

    $$v=v_0+at$$

    This one-dimensional form requires constant $a$ and a consistent positive direction.

11. Cartão 11

    Pergunta

    Where is an end-of-period cash flow placed?

    Resposta

    At the boundary marking the end of that period. Interest factors assume consistent period lengths and cash-flow timing.

12. Cartão 12

    Pergunta

    How is the FE reference handbook available during the exam?

    Resposta

    It is supplied onscreen as a searchable PDF. Practice locating information in the same current reference rather than relying on memory of a printed layout.

13. Cartão 13

    Pergunta

    Probability of $A$ or $B$ for possibly overlapping events?

    Resposta

    $$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

14. Cartão 14

    Pergunta

    SI base unit of time?

    Resposta

    The second, symbol $s$.

15. Cartão 15

    Pergunta

    Linear equation from one point and a slope?

    Resposta

    $$y-y_1=m(x-x_1)$$

    Use it when the line passes through $(x_1,y_1)$ and has slope $m$.

16. Cartão 16

    Pergunta

    Position after time $t$ under constant acceleration?

    Resposta

    $$x=x_0+v_0t+\frac{1}{2}at^2$$

    The acceleration is constant and all signed components use one coordinate direction.

17. Cartão 17

    Pergunta

    A project assignment lies outside your competence. Proper response?

    Resposta

    Accept only the parts you are qualified to perform, or obtain qualified supervision or support. Do not present unfamiliar work as your own competent judgment.

18. Cartão 18

    Pergunta

    SI base unit of electric current?

    Resposta

    The ampere, symbol $A$.

19. Cartão 19

    Pergunta

    Future amount under simple interest?

    Resposta

    $$F=P(1+in)$$

    Here $P$ is principal, $i$ is interest per period, and $n$ is the number of periods; interest is earned only on principal.

20. Cartão 20

    Pergunta

    Distance between two Cartesian points?

    Resposta

    $$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

    The coordinate components must use the same length unit.

21. Cartão 21

    Pergunta

    Constant-acceleration relation without elapsed time?

    Resposta

    $$v^2=v_0^2+2a(x-x_0)$$

    Use signed one-dimensional components and constant $a$.

22. Cartão 22

    Pergunta

    Where do you search the supplied reference material in the exam interface?

    Resposta

    Use the exam software's search box beside the supplied reference material.

23. Cartão 23

    Pergunta

    Conditional probability of $A$ given $B$?

    Resposta

    $$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$$

    This requires $P(B)>0$.

24. Cartão 24

    Pergunta

    SI base unit of thermodynamic temperature?

    Resposta

    The kelvin, symbol $K$. Kelvin is written without a degree sign.

25. Cartão 25

    Pergunta

    A client asks you to remove an unfavorable test result from a report. Proper response?

    Resposta

    Keep the material result and explain it accurately. Engineering statements should be objective, complete enough to avoid deception, and supported by evidence.

26. Cartão 26

    Pergunta

    Midpoint of two Cartesian points?

    Resposta

    $$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$

    Average corresponding coordinates.

27. Cartão 27

    Pergunta

    Linear momentum of a particle?

    Resposta

    $$\mathbf{p}=m\mathbf{v}$$

    For constant mass, SI units are $kg\cdot m/s$.

28. Cartão 28

    Pergunta

    Future amount under compound interest?

    Resposta

    $$F=P(1+i)^n$$

    The rate period must match the period used for $n$.

29. Cartão 29

    Pergunta

    Should an FE search workflow depend on Ctrl+F?

    Resposta

    No. The May 2026 NCEES Examinee Guide says Ctrl+F isn't available; practice with the exam interface's own search box.

30. Cartão 30

    Pergunta

    SI base unit of amount of substance?

    Resposta

    The mole, symbol $mol$.

31. Cartão 31

    Pergunta

    Joint probability expressed with a conditional probability?

    Resposta

    $$P(A\cap B)=P(A\mid B)P(B)$$

    This elementary formula requires $P(B)>0$.

32. Cartão 32

    Pergunta

    Standard equation of a circle centered at $(h,k)$?

    Resposta

    $$(x-h)^2+(y-k)^2=r^2$$

    The coordinate units and radius unit must match.

33. Cartão 33

    Pergunta

    Impulse-momentum relationship?

    Resposta

    $$\int_{t_1}^{t_2}\sum\mathbf{F}\,dt=\mathbf{p}_2-\mathbf{p}_1$$

    The force is the net external force on the chosen system.

34. Cartão 34

    Pergunta

    You discover a financial interest that could affect your recommendation. Proper response?

    Resposta

    Disclose the conflict promptly to affected parties and obtain appropriate consent or withdraw. Hidden personal interests undermine independent judgment.

35. Cartão 35

    Pergunta

    SI base unit of luminous intensity?

    Resposta

    The candela, symbol $cd$.

36. Cartão 36

    Pergunta

    Present worth of one future amount?

    Resposta

    $$P=\frac{F}{(1+i)^n}$$

    Use an effective rate $i$ per period consistent with $n$.

37. Cartão 37

    Pergunta

    Roots of $ax^2+bx+c=0$?

    Resposta

    $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

    This form requires $a\ne0$.

38. Cartão 38

    Pergunta

    When is total linear momentum conserved?

    Resposta

    When net external impulse on the system is zero. Internal forces can redistribute momentum but do not change the system total.

39. Cartão 39

    Pergunta

    Why avoid memorizing FE handbook page numbers?

    Resposta

    Pagination and layout can change. Remember section names, concepts, and strong search terms instead.

40. Cartão 40

    Pergunta

    Probability condition that defines independence of $A$ and $B$?

    Resposta

    $$P(A\cap B)=P(A)P(B)$$

    Independence is not the same as mutual exclusivity.

41. Cartão 41

    Pergunta

    SI unit of frequency in base-unit form?

    Resposta

    The hertz, $Hz=s^{-1}$, applies to periodic phenomena. One hertz is one cycle per second.

42. Cartão 42

    Pergunta

    Present worth of an end-of-period uniform series $A$?

    Resposta

    $$P=A\frac{(1+i)^n-1}{i(1+i)^n}$$

    This requires $i\ne0$; at $i=0$, $P=An$.

43. Cartão 43

    Pergunta

    What does $b^2-4ac$ reveal for a real-coefficient quadratic?

    Resposta

    It determines the root type: positive gives two distinct real roots, zero gives one repeated real root, and negative gives a complex-conjugate pair.

44. Cartão 44

    Pergunta

    Work done by a constant force through displacement?

    Resposta

    $$W=\mathbf{F}\cdot\mathbf{s}=Fs\cos\theta$$

    Use the angle between force and displacement; SI units are joules.

45. Cartão 45

    Pergunta

    A vendor offers a valuable gift while you evaluate bids. Proper response?

    Resposta

    Decline it and follow the organization's disclosure policy. Even an intended courtesy can create an actual or apparent conflict.

46. Cartão 46

    Pergunta

    SI unit of force in base-unit form?

    Resposta

    The newton, $N=kg\cdot m/s^2$.

47. Cartão 47

    Pergunta

    First search cue for a named quantity such as coefficient of thermal expansion?

    Resposta

    Search the exact technical noun phrase. Distinctive nouns usually narrow results better than a whole question sentence.

48. Cartão 48

    Pergunta

    Bayes' rule for $P(A\mid B)$?

    Resposta

    $$P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}$$

    This requires $P(B)>0$.

49. Cartão 49

    Pergunta

    Product of powers with the same positive base and real exponents?

    Resposta

    $$a^m a^n=a^{m+n}$$

    This form assumes $a>0$ and real exponents $m$ and $n$.

50. Cartão 50

    Pergunta

    Translational kinetic energy of a particle?

    Resposta

    $$K=\frac{1}{2}mv^2$$

    Mass and speed must use a coherent unit system.

51. Cartão 51

    Pergunta

    SI unit of pressure in force-area form?

    Resposta

    The pascal, $Pa=N/m^2$. It is the SI derived unit for pressure and stress.

52. Cartão 52

    Pergunta

    Equivalent annual amount for present worth $P$?

    Resposta

    $$A=P\frac{i(1+i)^n}{(1+i)^n-1}$$

    For the displayed formula, $i\ne0$; at $i=0$, $A=P/n$. Payments occur at period ends and $i$ matches that period.

53. Cartão 53

    Pergunta

    May confidential client information be shared for personal benefit?

    Resposta

    No. Protect it unless the client authorizes disclosure or law, safety, or professional duty requires it.

54. Cartão 54

    Pergunta

    A handbook search term returns too many results. Best refinement?

    Resposta

    Add one meaningful qualifier, such as the phenomenon, geometry, distribution, or material property you need.

55. Cartão 55

    Pergunta

    Quotient of powers with the same positive base and real exponents?

    Resposta

    $$\frac{a^m}{a^n}=a^{m-n}$$

    This form assumes $a>0$ and real exponents $m$ and $n$.

56. Cartão 56

    Pergunta

    Net work-energy relationship for a particle?

    Resposta

    $$W_{net}=K_2-K_1$$

    The net work includes the work of all forces represented in the particle model.

57. Cartão 57

    Pergunta

    Ordered selections of $r$ objects from $n$ distinct objects?

    Resposta

    $$P(n,r)=\frac{n!}{(n-r)!}$$

    Use this when order matters and selection is without replacement.

58. Cartão 58

    Pergunta

    Future worth of an end-of-period uniform series $A$?

    Resposta

    $$F=A\frac{(1+i)^n-1}{i}$$

    This requires $i\ne0$; at $i=0$, $F=An$.

59. Cartão 59

    Pergunta

    Power of a power for a positive base and real exponents?

    Resposta

    $$(a^m)^n=a^{mn}$$

    This form assumes $a>0$ and real exponents $m$ and $n$.

60. Cartão 60

    Pergunta

    SI unit of energy in force-distance form?

    Resposta

    The joule, $J=N\cdot m$.

61. Cartão 61

    Pergunta

    A problem gives only an unfamiliar symbol. Best search preparation?

    Resposta

    Translate the symbol into its likely quantity name and discipline context before searching; symbol-only searches are often ambiguous.

62. Cartão 62

    Pergunta

    Confidential information reveals an imminent public-safety risk. What comes first?

    Resposta

    Take steps required to protect people, using appropriate legal and organizational channels. Share only what is necessary and document the basis for escalation.

63. Cartão 63

    Pergunta

    Unordered selections of $r$ objects from $n$ distinct objects?

    Resposta

    $$C(n,r)=\frac{n!}{r!(n-r)!}$$

    Use this when order does not matter and selection is without replacement.

64. Cartão 64

    Pergunta

    Change in gravitational potential energy near Earth's surface?

    Resposta

    $$\Delta U_g=mg\Delta h$$

    Use approximately constant $g$ and a clearly chosen height datum.

65. Cartão 65

    Pergunta

    Negative-exponent rule for a positive base and real exponent?

    Resposta

    $$a^{-n}=\frac{1}{a^n}$$

    This form assumes $a>0$ and real exponent $n$.

66. Cartão 66

    Pergunta

    Uniform end-of-period deposit needed for future amount $F$?

    Resposta

    $$A=F\frac{i}{(1+i)^n-1}$$

    For the displayed formula, $i\ne0$; at $i=0$, $A=F/n$. The first deposit is one period from now and the last occurs at period $n$.

67. Cartão 67

    Pergunta

    A handbook search for an acronym fails. Next move?

    Resposta

    Try the expanded term and a common alternate name. Acronyms vary by section and discipline.

68. Cartão 68

    Pergunta

    SI unit of power in energy-time form?

    Resposta

    The watt, $W=J/s$.

69. Cartão 69

    Pergunta

    Logarithm of a product?

    Resposta

    $$\log_b(xy)=\log_b x+\log_b y$$

    For real logarithms, $b>0$, $b\ne1$, and both arguments are positive.

70. Cartão 70

    Pergunta

    Conditions for a binomial model?

    Resposta

    A fixed number of independent trials, two outcomes per trial, and the same success probability on every trial.

71. Cartão 71

    Pergunta

    Using another engineer's analysis in your deliverable?

    Resposta

    Credit the source and verify the analysis before relying on it. Do not present someone else's work as your own.

72. Cartão 72

    Pergunta

    SI unit of electric charge in current-time form?

    Resposta

    The coulomb, $C=A\cdot s$.

73. Cartão 73

    Pergunta

    Average power from work over elapsed time?

    Resposta

    $$P_{avg}=\frac{W}{\Delta t}$$

    Use consistent energy and time units; watts are joules per second.

74. Cartão 74

    Pergunta

    Annual equivalent of cost $P$ with salvage $S$ after $n$ periods?

    Resposta

    $$A=P\frac{i(1+i)^n}{(1+i)^n-1}-S\frac{i}{(1+i)^n-1}$$

    The first factor converts present cost to an annual amount; the second converts future salvage. Use the same effective rate and period basis. At $i=0$, $A=(P-S)/n$.

75. Cartão 75

    Pergunta

    Logarithm of a positive quantity raised to a power?

    Resposta

    $$\log_b(x^p)=p\log_b x$$

    For real logarithms, $x>0$, $b>0$, and $b\ne1$.

76. Cartão 76

    Pergunta

    SI unit of electric potential difference in energy-charge form?

    Resposta

    The volt, $V=J/C$.

77. Cartão 77

    Pergunta

    Instantaneous mechanical power delivered by a force?

    Resposta

    $$P=\mathbf{F}\cdot\mathbf{v}$$

    Use the velocity of the force's point of application.

78. Cartão 78

    Pergunta

    A precise handbook term gives no useful match. Next move?

    Resposta

    Try a standard synonym or the broader parent concept, then narrow within the relevant section.

79. Cartão 79

    Pergunta

    Mean of $X\sim\operatorname{Binomial}(n,p)$?

    Resposta

    $$E[X]=np$$

80. Cartão 80

    Pergunta

    SI unit of electrical resistance in voltage-current form?

    Resposta

    The ohm, $\Omega=V/A$.

81. Cartão 81

    Pergunta

    Moment of a force about a point?

    Resposta

    $$\mathbf{M}=\mathbf{r}\times\mathbf{F}$$

    Vector $\mathbf{r}$ runs from the moment point to a point on the force's line of action.

82. Cartão 82

    Pergunta

    Cash-flow pattern described by an arithmetic gradient $G$?

    Resposta

    A series that changes by the same amount $G$ each period. In the usual convention, the first gradient increment appears after the first base payment.

83. Cartão 83

    Pergunta

    Inverse relationship between the natural log and exponential?

    Resposta

    $$\ln(e^x)=x,\qquad e^{\ln x}=x$$

    The second identity requires $x>0$.

84. Cartão 84

    Pergunta

    SI prefix giga?

    Resposta

    $G=10^9$.

85. Cartão 85

    Pergunta

    Angular velocity after time $t$ under constant angular acceleration?

    Resposta

    $$\omega=\omega_0+\alpha t$$

    Use consistent angular sign convention and radians for angular measures.

86. Cartão 86

    Pergunta

    When should an engineer sign or seal technical work?

    Resposta

    Only when authorized and when the work was prepared by the engineer or under the level of responsible charge required by the applicable jurisdiction.

87. Cartão 87

    Pergunta

    Best memory target for returning to a handbook formula?

    Resposta

    The section or concept name plus a distinctive search phrase. That survives layout changes better than a page number.

88. Cartão 88

    Pergunta

    SI prefix mega?

    Resposta

    $M=10^6$. Uppercase matters: lowercase $m$ means milli.

89. Cartão 89

    Pergunta

    Convert degrees to radians?

    Resposta

    $$\theta_{rad}=\theta_{deg}\frac{\pi}{180}$$

    Use radians in calculus formulas unless a formula explicitly says otherwise.

90. Cartão 90

    Pergunta

    Variance of $X\sim\operatorname{Binomial}(n,p)$?

    Resposta

    $$\operatorname{Var}(X)=np(1-p)$$

91. Cartão 91

    Pergunta

    Cash-flow pattern described by a geometric gradient rate $g$?

    Resposta

    A series in which each payment is the previous payment multiplied by $1+g$. Keep $g$ on the same period basis as the interest rate.

92. Cartão 92

    Pergunta

    Where do licensure and sealing requirements come from?

    Resposta

    The applicable jurisdiction's current laws and board rules. Confirm local requirements rather than assuming one state's rules apply everywhere.

93. Cartão 93

    Pergunta

    Rotational kinetic energy about a fixed axis?

    Resposta

    $$K_{rot}=\frac{1}{2}I\omega^2$$

    $I$ is the mass moment of inertia about that axis.

94. Cartão 94

    Pergunta

    SI prefix kilo?

    Resposta

    $k=10^3$.

95. Cartão 95

    Pergunta

    Right-triangle definition of sine?

    Resposta

    Sine is opposite over hypotenuse. This definition applies to an acute angle in a right triangle.

96. Cartão 96

    Pergunta

    After search lands near a result, what should you scan first?

    Resposta

    The section heading and nearby subheading. They establish whether the result belongs to the right model and subject.

97. Cartão 97

    Pergunta

    Rotational analogue of Newton's second law about a fixed principal axis?

    Resposta

    $$\sum M=I\alpha$$

    This scalar form assumes the stated fixed-axis model and a consistent sign convention.

98. Cartão 98

    Pergunta

    SI prefix milli?

    Resposta

    $m=10^{-3}$. Context distinguishes the prefix from the metre symbol.

99. Cartão 99

    Pergunta

    Meaning of a nominal annual rate compounded $m$ times per year?

    Resposta

    It is a quoted annual rate equal to $m$ times the periodic rate. It is not itself the effective annual rate unless compounding is annual.

100. Cartão 100

     Pergunta

     Standardize an observation $x$ from a distribution with mean $\mu$ and standard deviation $\sigma$?

     Resposta

     $$z=\frac{x-\mu}{\sigma}$$

     This requires $\sigma>0$; using a normal probability table also assumes an appropriate normal model.

101. Cartão 101

     Pergunta

     Right-triangle definition of cosine?

     Resposta

     Cosine is adjacent over hypotenuse. This definition applies to an acute angle in a right triangle.

102. Cartão 102

     Pergunta

     Force-displacement relation for an ideal linear spring?

     Resposta

     $$F=-kx$$

     The minus sign shows restoring direction; the model applies within the spring's linear range.

103. Cartão 103

     Pergunta

     May qualifications or project responsibility be exaggerated in a proposal?

     Resposta

     No. State education, licensure, experience, and responsibility accurately, including the actual roles of collaborators.

104. Cartão 104

     Pergunta

     SI prefix micro?

     Resposta

     $\mu=10^{-6}$.

105. Cartão 105

     Pergunta

     A handbook symbol conflicts with your usual notation. What controls?

     Resposta

     The definition beside the selected relationship. Symbols can represent different quantities in different sections.

106. Cartão 106

     Pergunta

     Periodic rate from nominal annual rate $r$ compounded $m$ times per year?

     Resposta

     $$i=\frac{r}{m}$$

     Use consistent decimal units, and pair this rate with the number of compounding periods.

107. Cartão 107

     Pergunta

     Right-triangle definition of tangent?

     Resposta

     Tangent is opposite over adjacent, or $\sin\theta/\cos\theta$ where $\cos\theta\ne0$.

108. Cartão 108

     Pergunta

     Average normal stress under centered axial load?

     Resposta

     $$\sigma=\frac{F}{A}$$

     $F$ is normal to area $A$; the average ignores local stress concentrations.

109. Cartão 109

     Pergunta

     Mean, median, and mode of a symmetric normal distribution?

     Resposta

     They are equal and located at $\mu$. The distribution is symmetric about that center.

110. Cartão 110

     Pergunta

     SI prefix nano?

     Resposta

     $n=10^{-9}$.

111. Cartão 111

     Pergunta

     Fundamental Pythagorean trigonometric identity?

     Resposta

     $$\sin^2\theta+\cos^2\theta=1$$

112. Cartão 112

     Pergunta

     Engineering normal strain?

     Resposta

     $$\varepsilon=\frac{\Delta L}{L_0}$$

     It is dimensionless and uses original gauge length $L_0$.

113. Cartão 113

     Pergunta

     A candidate retrieves a formula from the handbook. What must be checked with it?

     Resposta

     Its applicability assumptions and variable definitions. A familiar equation used under the wrong model is still wrong.

114. Cartão 114

     Pergunta

     Technical duty to an employer or client?

     Resposta

     Act as a faithful professional within lawful, ethical limits, communicate material risks, and preserve independent technical judgment.

115. Cartão 115

     Pergunta

     How does a length conversion factor apply to area?

     Resposta

     Square it. If $1\,u=a\,v$, then $1\,u^2=a^2\,v^2$.

116. Cartão 116

     Pergunta

     Effective annual rate from nominal rate $r$ compounded $m$ times per year?

     Resposta

     $$i_{eff}=\left(1+\frac{r}{m}\right)^m-1$$

117. Cartão 117

     Pergunta

     When is a Poisson count model reasonable?

     Resposta

     For counts in a fixed interval when events occur independently at an approximately constant average rate and simultaneous events are negligible.

118. Cartão 118

     Pergunta

     Relationship for the third side of a non-right triangle?

     Resposta

     $$c^2=a^2+b^2-2ab\cos C$$

     Angle $C$ lies between sides $a$ and $b$, opposite side $c$.

119. Cartão 119

     Pergunta

     Linear elastic stress-strain relation in uniaxial loading?

     Resposta

     $$\sigma=E\varepsilon$$

     Use it in the material's linear elastic range; $E$ has stress units.

120. Cartão 120

     Pergunta

     How does a length conversion factor apply to volume?

     Resposta

     Cube it. If $1\,u=a\,v$, then $1\,u^3=a^3\,v^3$.

121. Cartão 121

     Pergunta

     What should you record for every variable in a retrieved equation?

     Resposta

     Its meaning and expected unit or dimension. This catches swapped symbols and incompatible substitutions.

122. Cartão 122

     Pergunta

     Effective annual rate for continuous compounding at nominal rate $r$?

     Resposta

     $$i_{eff}=e^r-1$$

     Here $r$ is expressed as a decimal per year.

123. Cartão 123

     Pergunta

     A supervisor directs an illegal engineering action. Proper response?

     Resposta

     Refuse the illegal action, document the direction, and escalate through appropriate legal or organizational channels.

124. Cartão 124

     Pergunta

     Magnitude of a three-component vector $\mathbf{v}$?

     Resposta

     $$\lVert\mathbf{v}\rVert=\sqrt{v_x^2+v_y^2+v_z^2}$$

     All components must represent the same physical quantity in compatible units.

125. Cartão 125

     Pergunta

     Mass density?

     Resposta

     $$\rho=\frac{m}{V}$$

     Use mass, not weight, in the numerator.

126. Cartão 126

     Pergunta

     Inches in one foot?

     Resposta

     $12\,in=1\,ft$.

127. Cartão 127

     Pergunta

     Mean and variance of $X\sim\operatorname{Poisson}(\lambda)$?

     Resposta

     Both equal $\lambda$. The standard deviation is $\sqrt{\lambda}$.

128. Cartão 128

     Pergunta

     How should subscripts be handled in a retrieved relationship?

     Resposta

     Read what each subscript identifies—state, direction, phase, location, or reference—before substituting values.

129. Cartão 129

     Pergunta

     Dot product from vector magnitudes and included angle?

     Resposta

     $$\mathbf{a}\cdot\mathbf{b}=\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert\cos\theta$$

     The result is a scalar.

130. Cartão 130

     Pergunta

     Average normal pressure on an area?

     Resposta

     $$p=\frac{F_n}{A}$$

     $F_n$ is the normal force; pressure is scalar in a fluid at rest.

131. Cartão 131

     Pergunta

     Feet in one yard?

     Resposta

     $3\,ft=1\,yd$.

132. Cartão 132

     Pergunta

     When are two interest rates equivalent?

     Resposta

     When they produce the same accumulation over the same time interval. Convert both rates to a common effective period before comparing them.

133. Cartão 133

     Pergunta

     You discover a material error in issued work. Proper response?

     Resposta

     Notify responsible parties promptly, assess the safety impact, correct the record, and document the response. Do not hide the error.

134. Cartão 134

     Pergunta

     Expected value of a discrete random variable?

     Resposta

     $$E[X]=\sum_x xP(X=x)$$

     Sum over all possible values with a valid probability mass function.

135. Cartão 135

     Pergunta

     Dot-product test for perpendicular nonzero vectors?

     Resposta

     Their dot product is zero. The zero vector also has zero dot product with every vector, so require both vectors to be nonzero before concluding perpendicularity.

136. Cartão 136

     Pergunta

     Hydrostatic pressure change with vertical depth in a constant-density fluid?

     Resposta

     $$\Delta p=\rho g\Delta h$$

     Depth increases downward and the fluid is at rest.

137. Cartão 137

     Pergunta

     Feet in one statute mile?

     Resposta

     $5280\,ft=1\,mi$.

138. Cartão 138

     Pergunta

     Present-worth acceptance rule for an independent investment at the minimum attractive rate of return?

     Resposta

     Accept it when net present worth is nonnegative. All cash flows must be discounted to the same time using the chosen comparison rate.

     

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139. Cartão 139

     Pergunta

     A retrieved formula contains signed quantities. What should you confirm?

     Resposta

     The stated sign convention and positive direction. Do not mix magnitudes with signed components silently.

140. Cartão 140

     Pergunta

     Magnitude of a vector cross product?

     Resposta

     $$\lVert\mathbf{a}\times\mathbf{b}\rVert=\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert\sin\theta$$

     The direction follows the right-hand rule.

141. Cartão 141

     Pergunta

     Exact inch-to-metre relationship?

     Resposta

     $1\,in=0.0254\,m$ exactly.

142. Cartão 142

     Pergunta

     Volumetric flow rate through a cross-section?

     Resposta

     $$Q=Av_{avg}$$

     Velocity is the area-average component normal to the section.

143. Cartão 143

     Pergunta

     Variance from the mean $\mu$?

     Resposta

     $$\operatorname{Var}(X)=E[(X-\mu)^2]$$

     Variance has squared units; standard deviation has the original units.

144. Cartão 144

     Pergunta

     Future-worth acceptance rule for an independent investment?

     Resposta

     Accept it when net future worth at the study horizon is nonnegative, using the minimum attractive rate of return consistently.

145. Cartão 145

     Pergunta

     What coordinate detail should be checked before using a vector or mechanics formula?

     Resposta

     Axis directions and the reference point or origin. Components and moments depend on those choices.

146. Cartão 146

     Pergunta

     Exact foot-to-metre relationship?

     Resposta

     $1\,ft=0.3048\,m$ exactly.

147. Cartão 147

     Pergunta

     How should important uncertainty be communicated?

     Resposta

     State the assumptions, limits, and plausible consequences clearly enough for the decision maker to understand the risk.

148. Cartão 148

     Pergunta

     When is the matrix product $AB$ defined?

     Resposta

     It is defined when the number of columns of $A$ equals the number of rows of $B$. If $A$ is $m\times n$ and $B$ is $n\times p$, then $AB$ is $m\times p$.

149. Cartão 149

     Pergunta

     Steady one-inlet, one-outlet mass continuity?

     Resposta

     $$\rho_1A_1v_1=\rho_2A_2v_2$$

     Here $v_1$ and $v_2$ are cross-sectional average velocity components normal to their sections, with density treated as uniform over each section. For incompressible flow with constant density, this reduces to $A_1v_1=A_2v_2$.

150. Cartão 150

     Pergunta

     Sample mean of observations $x_1,\ldots,x_n$?

     Resposta

     $$\bar{x}=\frac{1}{n}\sum_{i=1}^n x_i$$

151. Cartão 151

     Pergunta

     Square feet in one acre?

     Resposta

     $1\,acre=43{,}560\,ft^2$.

152. Cartão 152

     Pergunta

     Determinant of $\begin{bmatrix}a&b\\c&d\end{bmatrix}$?

     Resposta

     $$ad-bc$$

153. Cartão 153

     Pergunta

     Bernoulli relationship along a streamline for ideal steady flow?

     Resposta

     $$\frac{p}{\rho g}+\frac{v^2}{2g}+z=\text{constant}$$

     This basic form assumes steady, incompressible, negligible-viscosity flow with no shaft work or heat effects between points.

154. Cartão 154

     Pergunta

     May inconvenient measurements be discarded without a technical basis?

     Resposta

     No. Use a documented, technically justified rule for excluding data and preserve the original record.

155. Cartão 155

     Pergunta

     Annual-worth acceptance rule for an independent investment?

     Resposta

     Accept it when equivalent annual worth is nonnegative. Compare alternatives over compatible service assumptions.

156. Cartão 156

     Pergunta

     Before using a handbook equation, what unit-system check matters?

     Resposta

     Confirm that every input and any embedded constant use one compatible unit system. Convert first when necessary.

157. Cartão 157

     Pergunta

     Cubic metres in one litre?

     Resposta

     $1\,L=10^{-3}\,m^3$.

158. Cartão 158

     Pergunta

     What does a zero determinant mean for a square matrix?

     Resposta

     The matrix is singular and has no inverse. A corresponding linear system cannot have one unique solution.

159. Cartão 159

     Pergunta

     Reynolds number for internal flow using characteristic length $L$?

     Resposta

     $$Re=\frac{\rho vL}{\mu}=\frac{vL}{\nu}$$

     It is dimensionless; use consistent properties and the geometry-appropriate characteristic length.

160. Cartão 160

     Pergunta

     Unbiased sample variance for an independent sample from one population?

     Resposta

     $$s^2=\frac{1}{n-1}\sum_{i=1}^n(x_i-\bar{x})^2$$

     This requires at least two observations, so $n\ge2$. The $n-1$ denominator estimates population variance after fitting the sample mean.

161. Cartão 161

     Pergunta

     How should mutually exclusive alternatives be compared?

     Resposta

     Use one consistent economic measure over a common study period and select the economically preferred feasible alternative; do not accept every alternative with a positive individual worth.

162. Cartão 162

     Pergunta

     Cubic centimetres in one millilitre?

     Resposta

     $1\,mL=1\,cm^3$.

163. Cartão 163

     Pergunta

     Why keep contemporaneous engineering records?

     Resposta

     They preserve assumptions, decisions, calculations, approvals, and changes so the work can be checked and responsibilities understood.

164. Cartão 164

     Pergunta

     Fast check after substituting into a handbook formula?

     Resposta

     Verify that the resulting dimensions match the requested quantity and that additive terms share dimensions.

165. Cartão 165

     Pergunta

     Matrix solution of $A\mathbf{x}=\mathbf{b}$ when $A$ is invertible?

     Resposta

     $$\mathbf{x}=A^{-1}\mathbf{b}$$

     In numerical work, direct factorization is usually preferable to explicitly forming the inverse.

166. Cartão 166

     Pergunta

     Sensible heat for constant specific heat and no phase change?

     Resposta

     $$Q=mc\Delta T$$

     $c$ must match the material, temperature range, mass unit, and temperature-difference unit.

167. Cartão 167

     Pergunta

     Standard error of the mean for an i.i.d. sample with population standard deviation $\sigma$?

     Resposta

     $$SE(\bar{x})=\frac{\sigma}{\sqrt{n}}$$

     The observations are independent and identically distributed with common variance $\sigma^2$. Use $s$ for unknown $\sigma$ only when the selected inference method allows it.

168. Cartão 168

     Pergunta

     Safe setup for converting $25\,ft^2$ to square metres?

     Resposta

     $$25\,ft^2\left(\frac{0.3048\,m}{1\,ft}\right)^2$$

     The squared length factor cancels $ft^2$ and leaves $m^2$.

169. Cartão 169

     Pergunta

     Purpose of incremental analysis for mutually exclusive alternatives?

     Resposta

     It tests whether the additional investment in the higher-cost option earns the required return compared with the lower-cost option.

170. Cartão 170

     Pergunta

     Physical meaning of $dy/dx$?

     Resposta

     It is the instantaneous rate of change of $y$ with respect to $x$, with units of the $y$ unit divided by the $x$ unit.

171. Cartão 171

     Pergunta

     One-dimensional steady conduction through a plane wall?

     Resposta

     $$\dot Q=-kA\frac{dT}{dx}$$

     The minus sign indicates heat flows toward lower temperature; $k$ is thermal conductivity.

172. Cartão 172

     Pergunta

     When should you algebraically rearrange a handbook relationship?

     Resposta

     After confirming the correct equation, definitions, assumptions, and sign convention. Then isolate the requested variable without changing its domain restrictions.

173. Cartão 173

     Pergunta

     Safe setup for converting $2\,in^3$ to cubic metres?

     Resposta

     $$2\,in^3\left(\frac{0.0254\,m}{1\,in}\right)^3$$

     Cube the exact length conversion because the original quantity is volume.

174. Cartão 174

     Pergunta

     A client changes a safety-critical design requirement verbally. Proper response?

     Resposta

     Confirm the change in writing, analyze its consequences, and obtain the required approvals before relying on it.

175. Cartão 175

     Pergunta

     Derivative of $x^n$ for constant $n$?

     Resposta

     $$\frac{d}{dx}x^n=nx^{n-1}$$

     Apply domain restrictions when $n$ is non-integer or negative.

176. Cartão 176

     Pergunta

     Convective heat-transfer rate at a surface?

     Resposta

     $$\dot Q=hA(T_s-T_\infty)$$

     $h$ depends on the flow and geometry; the sign follows the chosen temperature difference.

177. Cartão 177

     Pergunta

     What does the central limit theorem say about a sample mean?

     Resposta

     For sufficiently large independent, identically distributed samples with finite variance, the sampling distribution of the properly standardized mean approaches normality. It does not make the raw data normal.

178. Cartão 178

     Pergunta

     Internal rate of return?

     Resposta

     The discount rate that makes net present worth zero. Multiple sign changes in cash flow can produce multiple or no meaningful real rates.

179. Cartão 179

     Pergunta

     Several similar handbook equations appear together. How do you choose?

     Resposta

     Match each equation's stated geometry, boundary conditions, material model, and variable definitions to the problem.

180. Cartão 180

     Pergunta

     Convert Celsius temperature to kelvin?

     Resposta

     $$T_K=T_{^\circ C}+273.15$$

     This conversion is for absolute temperature, not temperature difference.

181. Cartão 181

     Pergunta

     Derivative of a product $u(x)v(x)$?

     Resposta

     $$(uv)'=u'v+uv'$$

182. Cartão 182

     Pergunta

     Internal escalation does not resolve a serious public-safety threat. Next step?

     Resposta

     Use the next appropriate authority or reporting channel, consistent with law and the urgency of the hazard. Keep a factual record.

183. Cartão 183

     Pergunta

     Two-sided confidence interval for a population mean with known $\sigma$?

     Resposta

     $$\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$$

     Use the normal-model conditions and the critical value for confidence level $1-\alpha$.

184. Cartão 184

     Pergunta

     Net thermal radiation from a small gray surface to large surroundings?

     Resposta

     $$\dot Q=\varepsilon\sigma A(T_s^4-T_{sur}^4)$$

     Use absolute temperatures; the model assumes the stated gray-surface surroundings.

185. Cartão 185

     Pergunta

     Basic benefit-cost acceptance rule for an independent public project?

     Resposta

     Accept when the consistently defined benefit-cost ratio is at least one. For mutually exclusive choices, use an incremental comparison rather than ranking by individual ratios alone.

186. Cartão 186

     Pergunta

     Derivative of a composite function $f(g(x))$?

     Resposta

     $$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$$

187. Cartão 187

     Pergunta

     Why inspect boundary or initial conditions near a formula?

     Resposta

     They determine which solution form applies and can eliminate constants or terms. A formula without its conditions may be unusable.

188. Cartão 188

     Pergunta

     Convert Celsius temperature to Fahrenheit?

     Resposta

     $$T_{^\circ F}=\frac{9}{5}T_{^\circ C}+32$$

189. Cartão 189

     Pergunta

     Ethical basis for expert technical testimony?

     Resposta

     Offer opinions only within your competence, disclose important assumptions and limitations, and distinguish facts from professional judgment.

190. Cartão 190

     Pergunta

     Meaning of a breakeven value?

     Resposta

     The value of a selected variable that makes two alternatives economically equivalent or makes net worth zero. Other assumptions are held fixed.

191. Cartão 191

     Pergunta

     Derivative of $e^x$?

     Resposta

     $$\frac{d}{dx}e^x=e^x$$

192. Cartão 192

     Pergunta

     First-law energy balance for a closed system with work positive out?

     Resposta

     $$\Delta E=Q-W$$

     $Q$ is heat into the system and $W$ is work done by the system; state a different sign convention before using it.

193. Cartão 193

     Pergunta

     Frequentist meaning of a 95% confidence procedure?

     Resposta

     Across repeated samples, about 95% of intervals built by the procedure would contain the fixed population parameter. It is not a 95% probability assigned to this already-computed interval.

194. Cartão 194

     Pergunta

     A needed value falls between handbook table entries. What is the cue?

     Resposta

     Check whether interpolation is appropriate, keep both axes and units straight, and avoid extrapolating unless the source supports it.

195. Cartão 195

     Pergunta

     Derivative of $\ln x$ over the real domain?

     Resposta

     $$\frac{d}{dx}\ln x=\frac{1}{x}$$

     This real-valued form requires $x>0$.

196. Cartão 196

     Pergunta

     Relationship between Celsius and kelvin temperature differences?

     Resposta

     $1\,K$ of temperature difference equals $1\,^{\circ}C$ of temperature difference. Do not add $273.15$ to a difference.

197. Cartão 197

     Pergunta

     How should credit for team engineering work be assigned?

     Resposta

     Accurately recognize each contributor's work and responsibility. Do not claim sole credit for a collaborative result.

198. Cartão 198

     Pergunta

     Main limitation of simple payback period?

     Resposta

     It ignores the time value of money and usually ignores cash flows after payback. Discounted payback fixes the first issue, not the second.

199. Cartão 199

     Pergunta

     Roles of the null and alternative hypotheses?

     Resposta

     The null states the baseline parameter claim tested; the alternative states the competing direction or difference supported when evidence is strong enough.

200. Cartão 200

     Pergunta

     Ideal-gas equation of state?

     Resposta

     $$pV=nRT$$

     Use absolute temperature and a gas constant consistent with pressure, volume, amount, and energy units.

201. Cartão 201

     Pergunta

     First check when reading a handbook table?

     Resposta

     Read the title, column headings, units, notes, and basis conditions before selecting a value.

202. Cartão 202

     Pergunta

     Candidates for absolute extrema of a continuous function on a closed interval?

     Resposta

     The endpoints and every interior point where the first derivative is zero or does not exist. Evaluate the function at each candidate; the largest and smallest values are the absolute extrema.

203. Cartão 203

     Pergunta

     How should a sunk cost affect a current engineering-economy decision?

     Resposta

     It should not affect the choice because it cannot be changed by the decision. Use future incremental cash flows instead.

204. Cartão 204

     Pergunta

     Convert Fahrenheit absolute temperature to Rankine?

     Resposta

     $$T_{^\circ R}=T_{^\circ F}+459.67$$

     Rankine uses the Fahrenheit-sized degree from absolute zero.

205. Cartão 205

     Pergunta

     Ohm's law for an ideal resistor?

     Resposta

     $$V=IR$$

     Use a consistent passive sign convention; $R$ is constant only over the assumed operating range.

206. Cartão 206

     Pergunta

     You have confidential details about a competitor's bid. Proper response?

     Resposta

     Do not use or disclose improperly obtained information. Notify the appropriate procurement or ethics contact.

207. Cartão 207

     Pergunta

     Physical meaning of a definite integral $\int_a^b f(x)\,dx$?

     Resposta

     It is signed accumulation of $f$ from $a$ to $b$. Its units are the units of $f$ multiplied by the units of $x$.

208. Cartão 208

     Pergunta

     Is a radian dimensionless?

     Resposta

     Yes. It is the ratio of arc length to radius, so the length units cancel; keep $rad$ when it clarifies angular meaning.

209. Cartão 209

     Pergunta

     Electrical power absorbed under the passive sign convention?

     Resposta

     $$P=VI$$

     For a resistor, this also gives $P=I^2R=V^2/R$ when Ohm's law applies.

210. Cartão 210

     Pergunta

     Meaning of a p-value?

     Resposta

     Assuming the null hypothesis and model are correct, it is the probability of a result at least as extreme as the observed result. It is not the probability that the null is true.

211. Cartão 211

     Pergunta

     First check when reading a handbook graph?

     Resposta

     Identify both axes, scales, units, curve labels, and any log scale before estimating a value.

212. Cartão 212

     Pergunta

     Difference between mass and weight?

     Resposta

     Mass measures inertia; weight is gravitational force on that mass. Their dimensions and units are different.

213. Cartão 213

     Pergunta

     Kirchhoff's current law at a node?

     Resposta

     The algebraic sum of currents at a node is zero. Choose one sign convention, such as currents entering positive, and apply it consistently.

214. Cartão 214

     Pergunta

     Evaluate $\int_a^b f(x)\,dx$ from an antiderivative $F$?

     Resposta

     $$\int_a^b f(x)\,dx=F(b)-F(a)$$

     This applies when $F'=f$ on the interval under the usual continuity conditions.

215. Cartão 215

     Pergunta

     Opportunity cost?

     Resposta

     The value of the best forgone alternative. Include it when choosing one use of a scarce resource prevents another valuable use.

216. Cartão 216

     Pergunta

     Weight from mass in a gravitational field?

     Resposta

     $$W=mg$$

     Use compatible units; $g$ is local gravitational acceleration, not a universal constant for every location.

217. Cartão 217

     Pergunta

     Kirchhoff's voltage law around a closed loop?

     Resposta

     The algebraic sum of voltage rises and drops around the loop is zero. Traverse the loop once with a consistent polarity convention.

218. Cartão 218

     Pergunta

     Decision rule using significance level $\alpha$?

     Resposta

     Reject the null when the p-value is at most $\alpha$; otherwise fail to reject it. Failing to reject does not prove the null.

219. Cartão 219

     Pergunta

     Can proprietary methods from a former employer be reused freely?

     Resposta

     Only if you have the legal right or permission to use them. General skill and experience may transfer; confidential proprietary material does not.

220. Cartão 220

     Pergunta

     Pound-force to newtons?

     Resposta

     $1\,lbf\approx4.44822\,N$. Use more digits only when the required precision justifies them.

221. Cartão 221

     Pergunta

     When does a definite integral equal ordinary geometric area?

     Resposta

     When the integrand is nonnegative over the interval. If it changes sign, the integral is signed area; integrate the absolute value or split the interval for total area.

222. Cartão 222

     Pergunta

     Charge-voltage relation for an ideal capacitor?

     Resposta

     $$q=CV$$

     $C$ is capacitance in farads; $V$ is the voltage across the capacitor under the chosen polarity.

223. Cartão 223

     Pergunta

     Searching for a material or fluid property: what qualifiers often matter?

     Resposta

     Material identity, phase, temperature, pressure, and composition. Use only the qualifiers relevant to that property source.

224. Cartão 224

     Pergunta

     Force produced when one slug accelerates at $1\,ft/s^2$?

     Resposta

     $1\,lbf$. Equivalently, $1\,lbf=1\,slug\cdot ft/s^2$.

225. Cartão 225

     Pergunta

     Relationship among nominal rate $i$, real rate $r$, and inflation $f$?

     Resposta

     $$1+i=(1+r)(1+f)$$

     Keep cash-flow dollars and discount rates on the same real-or-nominal basis.

226. Cartão 226

     Pergunta

     Antiderivative of $x^n$ for $n\ne-1$?

     Resposta

     $$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$$

227. Cartão 227

     Pergunta

     Time constant of a first-order resistor-capacitor circuit?

     Resposta

     $$\tau=RC$$

     Use the resistance seen by the capacitor in the applicable first-order linear circuit.

228. Cartão 228

     Pergunta

     What physical quantity does pound-mass measure?

     Resposta

     Mass. Do not substitute $lbm$ directly where a force unit such as $lbf$ is required.

229. Cartão 229

     Pergunta

     Type I error?

     Resposta

     Rejecting a true null hypothesis. Its probability is controlled by the chosen significance level under the test assumptions.

230. Cartão 230

     Pergunta

     A search hit contains the right words. Is that enough?

     Resposta

     No. Read the surrounding definition, equation label, units, and notes to confirm the hit answers the actual question.

231. Cartão 231

     Pergunta

     Annual straight-line depreciation from basis $B$ to salvage $S$ over $n$ years?

     Resposta

     $$D=\frac{B-S}{n}$$

     This is an accounting allocation, not a market-value forecast.

232. Cartão 232

     Pergunta

     Purpose of $g_c$ in some US customary force-mass equations?

     Resposta

     It reconciles chosen $lbm$, $lbf$, foot, and second units. Include it only when the equation and unit convention require it; coherent SI equations do not use it.

233. Cartão 233

     Pergunta

     Recognizable form of a separable first-order differential equation?

     Resposta

     $$\frac{dy}{dx}=g(x)h(y)$$

     Rearrange to place the $y$ terms with $dy$ and the $x$ terms with $dx$, while checking for solutions lost by division.

234. Cartão 234

     Pergunta

     Using unverified software output in a safety-relevant calculation?

     Resposta

     Check inputs, assumptions, units, and results with an independent method appropriate to the risk. The engineer remains responsible for professional judgment.

235. Cartão 235

     Pergunta

     Equivalent resistance of ideal resistors in series?

     Resposta

     $$R_{eq}=\sum_i R_i$$

     Series elements carry the same current with no branching node between them.

236. Cartão 236

     Pergunta

     Pressure conversion from psi to pascals?

     Resposta

     $1\,psi\approx6894.76\,Pa$. Keep enough significant figures for the problem's data.

237. Cartão 237

     Pergunta

     Type II error?

     Resposta

     Failing to reject a false null hypothesis. Its probability is $\beta$ for a specified alternative.

238. Cartão 238

     Pergunta

     Useful flashcard to make after a failed handbook search?

     Resposta

     A short scenario-to-search-cue card recording the section name and terms that eventually worked—not a copied problem or page number.

239. Cartão 239

     Pergunta

     Standard form of a first-order linear differential equation?

     Resposta

     $$\frac{dy}{dx}+P(x)y=Q(x)$$

     An integrating factor is $e^{\int P(x)\,dx}$.

240. Cartão 240

     Pergunta

     Equivalent resistance of ideal resistors in parallel?

     Resposta

     $$\frac{1}{R_{eq}}=\sum_i\frac{1}{R_i}$$

     Parallel elements share the same two nodes and therefore the same voltage.

241. Cartão 241

     Pergunta

     Book value after $k$ years of straight-line depreciation?

     Resposta

     $$BV_k=B-kD$$

     Apply it within the depreciable life and do not depreciate below the assumed salvage value.

242. Cartão 242

     Pergunta

     Standard atmosphere in pascals?

     Resposta

     $1\,atm=101{,}325\,Pa$ exactly.

243. Cartão 243

     Pergunta

     Using AI-generated technical content in engineering work?

     Resposta

     Treat it as unverified input: protect confidential data, check sources and calculations, disclose use when required, and keep accountable human review.

244. Cartão 244

     Pergunta

     Trial form for a homogeneous linear ODE with constant coefficients?

     Resposta

     Use $y=e^{rx}$ to obtain the characteristic polynomial in $r$. Repeated or complex roots require the corresponding standard solution forms.

245. Cartão 245

     Pergunta

     Wave-speed relationship among frequency and wavelength?

     Resposta

     $$v=f\lambda$$

     Use frequency in cycles per second and wavelength in compatible length units.

246. Cartão 246

     Pergunta

     Statistical power?

     Resposta

     The probability of rejecting the null when a specified alternative is true: $1-\beta$. Larger effects, larger samples, or lower variability commonly increase power.

247. Cartão 247

     Pergunta

     Bar in pascals?

     Resposta

     $1\,bar=100{,}000\,Pa$ exactly.

248. Cartão 248

     Pergunta

     How should handbook navigation be practiced before the FE exam?

     Resposta

     Use the current handbook from MyNCEES while solving original or properly licensed practice problems, and rehearse the exact search terms that find definitions and assumptions.

249. Cartão 249

     Pergunta

     Is depreciation itself a cash outflow?

     Resposta

     No. It is a noncash accounting expense, though it can affect taxes and therefore after-tax cash flow.

250. Cartão 250

     Pergunta

     First-order linear approximation of $f(x)$ near $x=a$?

     Resposta

     $$f(x)\approx f(a)+f'(a)(x-a)$$

     It is most accurate for $x$ close to $a$ when higher derivatives stay bounded.

251. Cartão 251

     Pergunta

     Relationship between period and frequency?

     Resposta

     $$T=\frac{1}{f}$$

     This requires $f>0$; $T$ has time units.

252. Cartão 252

     Pergunta

     How should foreseeable environmental harm enter an engineering decision?

     Resposta

     Identify and communicate it, follow applicable requirements, and consider practicable ways to reduce harm without compromising public safety.

253. Cartão 253

     Pergunta

     Foot-pound-force in joules?

     Resposta

     $1\,ft\cdot lbf\approx1.35582\,J$. This is a work or energy conversion; keep torque labeled as force times distance even though its dimensions match energy.

254. Cartão 254

     Pergunta

     Allowed range and sign meaning of Pearson correlation $r$?

     Resposta

     $-1\le r\le1$. The sign gives the direction of linear association, and magnitude measures its linear strength.

255. Cartão 255

     Pergunta

     When should you search instead of relying on recall?

     Resposta

     Search when a constant, coefficient, convention, specialized relationship, or applicability detail could change the result. Recall can start the route; the reference should confirm details.

256. Cartão 256

     Pergunta

     After-tax effect of a deductible expense $E$ at marginal tax rate $t$?

     Resposta

     The net cost is $E(1-t)$ under the simplified assumption that the deduction can be used fully in that period.

257. Cartão 257

     Pergunta

     Centered finite-difference approximation to $f'(x)$?

     Resposta

     $$f'(x)\approx\frac{f(x+h)-f(x-h)}{2h}$$

     For smooth $f$, truncation error is proportional to $h^2$ before roundoff dominates.

258. Cartão 258

     Pergunta

     General material balance on a control volume?

     Resposta

     $$\text{accumulation}=\text{in}-\text{out}+\text{generation}-\text{consumption}$$

     At steady state, accumulation is zero; generation and consumption depend on reactions or sources.

259. Cartão 259

     Pergunta

     Approximate joules in one International Table Btu?

     Resposta

     $1\,Btu_{IT}\approx1055.06\,J$. Identify the Btu convention when precision matters.

260. Cartão 260

     Pergunta

     What is the defender in a replacement analysis?

     Resposta

     The currently owned asset being compared with a proposed replacement. Its current market value is an opportunity cost; its original purchase price is sunk.

261. Cartão 261

     Pergunta

     Speaking publicly about a technical matter outside your expertise?

     Resposta

     Limit claims to what you can support, identify the boundary of your expertise, and avoid implying qualifications you do not have.

262. Cartão 262

     Pergunta

     Single-interval trapezoidal approximation to an integral?

     Resposta

     $$\int_a^b f(x)\,dx\approx\frac{b-a}{2}[f(a)+f(b)]$$

     It is exact for linear $f$.

263. Cartão 263

     Pergunta

     Energy efficiency of a device?

     Resposta

     $$\eta=\frac{\text{useful output energy}}{\text{input energy}}$$

     Use the same energy basis in numerator and denominator; a real passive device has $0\le\eta\le1$.

264. Cartão 264

     Pergunta

     Purpose of sensitivity analysis in engineering economics?

     Resposta

     It shows how the decision changes when an uncertain input changes. Vary inputs over plausible ranges while preserving units and model relationships.

265. Cartão 265

     Pergunta

     Can a strong correlation establish causation?

     Resposta

     No. Correlation alone cannot rule out confounding, reverse direction, selection effects, or coincidence.

266. Cartão 266

     Pergunta

     Where should you confirm the tested topics for your FE discipline?

     Resposta

     Use the discipline specification linked from the official NCEES FE exam hub. This common deck does not replace that discipline-specific boundary.

267. Cartão 267

     Pergunta

     Mechanical horsepower in watts?

     Resposta

     $1\,hp\approx745.7\,W$, based on $550\,ft\cdot lbf/s$. Distinguish it from metric horsepower.

268. Cartão 268

     Pergunta

     Combine independent standard uncertainties $u_1,\ldots,u_n$ in a sum or difference?

     Resposta

     $$u_c=\sqrt{u_1^2+u_2^2+\cdots+u_n^2}$$

     This root-sum-square form assumes uncorrelated inputs and coefficients of magnitude one; otherwise include sensitivities and covariance.

269. Cartão 269

     Pergunta

     Newton iteration for a root of $f(x)=0$?

     Resposta

     $$x_{k+1}=x_k-\frac{f(x_k)}{f'(x_k)}$$

     It requires $f'(x_k)\ne0$ and a suitable starting value; convergence is not guaranteed.

270. Cartão 270

     Pergunta

     Meaning of slope in simple linear regression of $y$ on $x$?

     Resposta

     It is the estimated change in mean response $y$ for a one-unit increase in $x$, under the fitted linear model. Its units are $y$ units per $x$ unit.

271. Cartão 271

     Pergunta

     Kilowatt-hour in joules?

     Resposta

     $1\,kW\cdot h=3.6\times10^6\,J$. It is energy, not power.

272. Cartão 272

     Pergunta

     Percent error relative to an accepted nonzero value?

     Resposta

     $$\%\,error=\left|\frac{x_{measured}-x_{accepted}}{x_{accepted}}\right|100\%$$

     The accepted value must be nonzero and both values must use compatible units.

273. Cartão 273

     Pergunta

     Does legal compliance alone settle an engineering ethics question?

     Resposta

     No. Law sets mandatory boundaries, while professional duties can require stronger action, especially when safety, honesty, or conflicts are involved.

274. Cartão 274

     Pergunta

     Does handbook fluency replace discipline-specific problem practice?

     Resposta

     No. Pair navigation and common recall with the current discipline specification and timed, rights-safe problems that match your selected FE discipline.

275. Cartão 275

     Pergunta

     Linear interpolation between $(x_1,y_1)$ and $(x_2,y_2)$?

     Resposta

     $$y\approx y_1+(y_2-y_1)\frac{x-x_1}{x_2-x_1}$$

     Use it between distinct $x$ values when local linear behavior is reasonable.

276. Cartão 276

     Pergunta

     Dimensional requirement for a physically meaningful additive equation?

     Resposta

     Every term being added or equated must have the same dimensions. A dimensionally consistent equation can still be physically wrong, but an inconsistent one is wrong.