FE Exam Flashcards: Engineering Equations, Units & Handbook Skills

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1. Carta 1

   Domanda

   SI base unit of length?

   Risposta

   The metre, symbol $m$.

2. Carta 2

   Domanda

   Slope between two points?

   Risposta

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

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

3. Carta 3

   Domanda

   Net force relationship for constant mass in an inertial frame?

   Risposta

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

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

4. Carta 4

   Domanda

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

   Risposta

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

5. Carta 5

   Domanda

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

   Risposta

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

6. Carta 6

   Domanda

   Probability of the complement of event $A$?

   Risposta

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

7. Carta 7

   Domanda

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

   Risposta

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

8. Carta 8

   Domanda

   SI base unit of mass?

   Risposta

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

9. Carta 9

   Domanda

   Linear equation from slope and intercept?

   Risposta

   $$y=mx+b$$

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

10. Carta 10

    Domanda

    Velocity after time $t$ under constant acceleration?

    Risposta

    $$v=v_0+at$$

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

11. Carta 11

    Domanda

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

    Risposta

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

12. Carta 12

    Domanda

    How is the FE reference handbook available during the exam?

    Risposta

    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. Carta 13

    Domanda

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

    Risposta

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

14. Carta 14

    Domanda

    SI base unit of time?

    Risposta

    The second, symbol $s$.

15. Carta 15

    Domanda

    Linear equation from one point and a slope?

    Risposta

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

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

16. Carta 16

    Domanda

    Position after time $t$ under constant acceleration?

    Risposta

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

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

17. Carta 17

    Domanda

    A project assignment lies outside your competence. Proper response?

    Risposta

    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. Carta 18

    Domanda

    SI base unit of electric current?

    Risposta

    The ampere, symbol $A$.

19. Carta 19

    Domanda

    Future amount under simple interest?

    Risposta

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

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

20. Carta 20

    Domanda

    Distance between two Cartesian points?

    Risposta

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

    The coordinate components must use the same length unit.

21. Carta 21

    Domanda

    Constant-acceleration relation without elapsed time?

    Risposta

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

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

22. Carta 22

    Domanda

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

    Risposta

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

23. Carta 23

    Domanda

    Conditional probability of $A$ given $B$?

    Risposta

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

    This requires $P(B)>0$.

24. Carta 24

    Domanda

    SI base unit of thermodynamic temperature?

    Risposta

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

25. Carta 25

    Domanda

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

    Risposta

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

26. Carta 26

    Domanda

    Midpoint of two Cartesian points?

    Risposta

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

    Average corresponding coordinates.

27. Carta 27

    Domanda

    Linear momentum of a particle?

    Risposta

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

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

28. Carta 28

    Domanda

    Future amount under compound interest?

    Risposta

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

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

29. Carta 29

    Domanda

    Should an FE search workflow depend on Ctrl+F?

    Risposta

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

30. Carta 30

    Domanda

    SI base unit of amount of substance?

    Risposta

    The mole, symbol $mol$.

31. Carta 31

    Domanda

    Joint probability expressed with a conditional probability?

    Risposta

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

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

32. Carta 32

    Domanda

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

    Risposta

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

    The coordinate units and radius unit must match.

33. Carta 33

    Domanda

    Impulse-momentum relationship?

    Risposta

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

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

34. Carta 34

    Domanda

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

    Risposta

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

35. Carta 35

    Domanda

    SI base unit of luminous intensity?

    Risposta

    The candela, symbol $cd$.

36. Carta 36

    Domanda

    Present worth of one future amount?

    Risposta

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

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

37. Carta 37

    Domanda

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

    Risposta

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

    This form requires $a\ne0$.

38. Carta 38

    Domanda

    When is total linear momentum conserved?

    Risposta

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

39. Carta 39

    Domanda

    Why avoid memorizing FE handbook page numbers?

    Risposta

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

40. Carta 40

    Domanda

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

    Risposta

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

    Independence is not the same as mutual exclusivity.

41. Carta 41

    Domanda

    SI unit of frequency in base-unit form?

    Risposta

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

42. Carta 42

    Domanda

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

    Risposta

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

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

43. Carta 43

    Domanda

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

    Risposta

    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. Carta 44

    Domanda

    Work done by a constant force through displacement?

    Risposta

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

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

45. Carta 45

    Domanda

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

    Risposta

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

46. Carta 46

    Domanda

    SI unit of force in base-unit form?

    Risposta

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

47. Carta 47

    Domanda

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

    Risposta

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

48. Carta 48

    Domanda

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

    Risposta

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

    This requires $P(B)>0$.

49. Carta 49

    Domanda

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

    Risposta

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

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

50. Carta 50

    Domanda

    Translational kinetic energy of a particle?

    Risposta

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

    Mass and speed must use a coherent unit system.

51. Carta 51

    Domanda

    SI unit of pressure in force-area form?

    Risposta

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

52. Carta 52

    Domanda

    Equivalent annual amount for present worth $P$?

    Risposta

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

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

53. Carta 53

    Domanda

    May confidential client information be shared for personal benefit?

    Risposta

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

54. Carta 54

    Domanda

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

    Risposta

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

55. Carta 55

    Domanda

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

    Risposta

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

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

56. Carta 56

    Domanda

    Net work-energy relationship for a particle?

    Risposta

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

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

57. Carta 57

    Domanda

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

    Risposta

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

    Use this when order matters and selection is without replacement.

58. Carta 58

    Domanda

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

    Risposta

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

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

59. Carta 59

    Domanda

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

    Risposta

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

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

60. Carta 60

    Domanda

    SI unit of energy in force-distance form?

    Risposta

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

61. Carta 61

    Domanda

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

    Risposta

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

62. Carta 62

    Domanda

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

    Risposta

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

63. Carta 63

    Domanda

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

    Risposta

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

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

64. Carta 64

    Domanda

    Change in gravitational potential energy near Earth's surface?

    Risposta

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

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

65. Carta 65

    Domanda

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

    Risposta

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

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

66. Carta 66

    Domanda

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

    Risposta

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

    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. Carta 67

    Domanda

    A handbook search for an acronym fails. Next move?

    Risposta

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

68. Carta 68

    Domanda

    SI unit of power in energy-time form?

    Risposta

    The watt, $W=J/s$.

69. Carta 69

    Domanda

    Logarithm of a product?

    Risposta

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

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

70. Carta 70

    Domanda

    Conditions for a binomial model?

    Risposta

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

71. Carta 71

    Domanda

    Using another engineer's analysis in your deliverable?

    Risposta

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

72. Carta 72

    Domanda

    SI unit of electric charge in current-time form?

    Risposta

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

73. Carta 73

    Domanda

    Average power from work over elapsed time?

    Risposta

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

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

74. Carta 74

    Domanda

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

    Risposta

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

    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. Carta 75

    Domanda

    Logarithm of a positive quantity raised to a power?

    Risposta

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

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

76. Carta 76

    Domanda

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

    Risposta

    The volt, $V=J/C$.

77. Carta 77

    Domanda

    Instantaneous mechanical power delivered by a force?

    Risposta

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

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

78. Carta 78

    Domanda

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

    Risposta

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

79. Carta 79

    Domanda

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

    Risposta

    $$E[X]=np$$

80. Carta 80

    Domanda

    SI unit of electrical resistance in voltage-current form?

    Risposta

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

81. Carta 81

    Domanda

    Moment of a force about a point?

    Risposta

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

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

82. Carta 82

    Domanda

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

    Risposta

    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. Carta 83

    Domanda

    Inverse relationship between the natural log and exponential?

    Risposta

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

    The second identity requires $x>0$.

84. Carta 84

    Domanda

    SI prefix giga?

    Risposta

    $G=10^9$.

85. Carta 85

    Domanda

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

    Risposta

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

    Use consistent angular sign convention and radians for angular measures.

86. Carta 86

    Domanda

    When should an engineer sign or seal technical work?

    Risposta

    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. Carta 87

    Domanda

    Best memory target for returning to a handbook formula?

    Risposta

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

88. Carta 88

    Domanda

    SI prefix mega?

    Risposta

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

89. Carta 89

    Domanda

    Convert degrees to radians?

    Risposta

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

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

90. Carta 90

    Domanda

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

    Risposta

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

91. Carta 91

    Domanda

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

    Risposta

    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. Carta 92

    Domanda

    Where do licensure and sealing requirements come from?

    Risposta

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

93. Carta 93

    Domanda

    Rotational kinetic energy about a fixed axis?

    Risposta

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

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

94. Carta 94

    Domanda

    SI prefix kilo?

    Risposta

    $k=10^3$.

95. Carta 95

    Domanda

    Right-triangle definition of sine?

    Risposta

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

96. Carta 96

    Domanda

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

    Risposta

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

97. Carta 97

    Domanda

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

    Risposta

    $$\sum M=I\alpha$$

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

98. Carta 98

    Domanda

    SI prefix milli?

    Risposta

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

99. Carta 99

    Domanda

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

    Risposta

    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. Carta 100

     Domanda

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

     Risposta

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

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

101. Carta 101

     Domanda

     Right-triangle definition of cosine?

     Risposta

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

102. Carta 102

     Domanda

     Force-displacement relation for an ideal linear spring?

     Risposta

     $$F=-kx$$

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

103. Carta 103

     Domanda

     May qualifications or project responsibility be exaggerated in a proposal?

     Risposta

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

104. Carta 104

     Domanda

     SI prefix micro?

     Risposta

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

105. Carta 105

     Domanda

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

     Risposta

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

106. Carta 106

     Domanda

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

     Risposta

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

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

107. Carta 107

     Domanda

     Right-triangle definition of tangent?

     Risposta

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

108. Carta 108

     Domanda

     Average normal stress under centered axial load?

     Risposta

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

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

109. Carta 109

     Domanda

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

     Risposta

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

110. Carta 110

     Domanda

     SI prefix nano?

     Risposta

     $n=10^{-9}$.

111. Carta 111

     Domanda

     Fundamental Pythagorean trigonometric identity?

     Risposta

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

112. Carta 112

     Domanda

     Engineering normal strain?

     Risposta

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

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

113. Carta 113

     Domanda

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

     Risposta

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

114. Carta 114

     Domanda

     Technical duty to an employer or client?

     Risposta

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

115. Carta 115

     Domanda

     How does a length conversion factor apply to area?

     Risposta

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

116. Carta 116

     Domanda

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

     Risposta

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

117. Carta 117

     Domanda

     When is a Poisson count model reasonable?

     Risposta

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

118. Carta 118

     Domanda

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

     Risposta

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

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

119. Carta 119

     Domanda

     Linear elastic stress-strain relation in uniaxial loading?

     Risposta

     $$\sigma=E\varepsilon$$

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

120. Carta 120

     Domanda

     How does a length conversion factor apply to volume?

     Risposta

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

121. Carta 121

     Domanda

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

     Risposta

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

122. Carta 122

     Domanda

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

     Risposta

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

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

123. Carta 123

     Domanda

     A supervisor directs an illegal engineering action. Proper response?

     Risposta

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

124. Carta 124

     Domanda

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

     Risposta

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

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

125. Carta 125

     Domanda

     Mass density?

     Risposta

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

     Use mass, not weight, in the numerator.

126. Carta 126

     Domanda

     Inches in one foot?

     Risposta

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

127. Carta 127

     Domanda

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

     Risposta

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

128. Carta 128

     Domanda

     How should subscripts be handled in a retrieved relationship?

     Risposta

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

129. Carta 129

     Domanda

     Dot product from vector magnitudes and included angle?

     Risposta

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

     The result is a scalar.

130. Carta 130

     Domanda

     Average normal pressure on an area?

     Risposta

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

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

131. Carta 131

     Domanda

     Feet in one yard?

     Risposta

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

132. Carta 132

     Domanda

     When are two interest rates equivalent?

     Risposta

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

133. Carta 133

     Domanda

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

     Risposta

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

134. Carta 134

     Domanda

     Expected value of a discrete random variable?

     Risposta

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

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

135. Carta 135

     Domanda

     Dot-product test for perpendicular nonzero vectors?

     Risposta

     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. Carta 136

     Domanda

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

     Risposta

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

     Depth increases downward and the fluid is at rest.

137. Carta 137

     Domanda

     Feet in one statute mile?

     Risposta

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

138. Carta 138

     Domanda

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

     Risposta

     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. Carta 139

     Domanda

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

     Risposta

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

140. Carta 140

     Domanda

     Magnitude of a vector cross product?

     Risposta

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

     The direction follows the right-hand rule.

141. Carta 141

     Domanda

     Exact inch-to-metre relationship?

     Risposta

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

142. Carta 142

     Domanda

     Volumetric flow rate through a cross-section?

     Risposta

     $$Q=Av_{avg}$$

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

143. Carta 143

     Domanda

     Variance from the mean $\mu$?

     Risposta

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

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

144. Carta 144

     Domanda

     Future-worth acceptance rule for an independent investment?

     Risposta

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

145. Carta 145

     Domanda

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

     Risposta

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

146. Carta 146

     Domanda

     Exact foot-to-metre relationship?

     Risposta

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

147. Carta 147

     Domanda

     How should important uncertainty be communicated?

     Risposta

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

148. Carta 148

     Domanda

     When is the matrix product $AB$ defined?

     Risposta

     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. Carta 149

     Domanda

     Steady one-inlet, one-outlet mass continuity?

     Risposta

     $$\rho_1A_1v_1=\rho_2A_2v_2$$

     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. Carta 150

     Domanda

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

     Risposta

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

151. Carta 151

     Domanda

     Square feet in one acre?

     Risposta

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

152. Carta 152

     Domanda

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

     Risposta

     $$ad-bc$$

153. Carta 153

     Domanda

     Bernoulli relationship along a streamline for ideal steady flow?

     Risposta

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

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

154. Carta 154

     Domanda

     May inconvenient measurements be discarded without a technical basis?

     Risposta

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

155. Carta 155

     Domanda

     Annual-worth acceptance rule for an independent investment?

     Risposta

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

156. Carta 156

     Domanda

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

     Risposta

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

157. Carta 157

     Domanda

     Cubic metres in one litre?

     Risposta

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

158. Carta 158

     Domanda

     What does a zero determinant mean for a square matrix?

     Risposta

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

159. Carta 159

     Domanda

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

     Risposta

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

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

160. Carta 160

     Domanda

     Unbiased sample variance for an independent sample from one population?

     Risposta

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

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

161. Carta 161

     Domanda

     How should mutually exclusive alternatives be compared?

     Risposta

     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. Carta 162

     Domanda

     Cubic centimetres in one millilitre?

     Risposta

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

163. Carta 163

     Domanda

     Why keep contemporaneous engineering records?

     Risposta

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

164. Carta 164

     Domanda

     Fast check after substituting into a handbook formula?

     Risposta

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

165. Carta 165

     Domanda

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

     Risposta

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

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

166. Carta 166

     Domanda

     Sensible heat for constant specific heat and no phase change?

     Risposta

     $$Q=mc\Delta T$$

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

167. Carta 167

     Domanda

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

     Risposta

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

     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. Carta 168

     Domanda

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

     Risposta

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

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

169. Carta 169

     Domanda

     Purpose of incremental analysis for mutually exclusive alternatives?

     Risposta

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

170. Carta 170

     Domanda

     Physical meaning of $dy/dx$?

     Risposta

     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. Carta 171

     Domanda

     One-dimensional steady conduction through a plane wall?

     Risposta

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

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

172. Carta 172

     Domanda

     When should you algebraically rearrange a handbook relationship?

     Risposta

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

173. Carta 173

     Domanda

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

     Risposta

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

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

174. Carta 174

     Domanda

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

     Risposta

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

175. Carta 175

     Domanda

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

     Risposta

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

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

176. Carta 176

     Domanda

     Convective heat-transfer rate at a surface?

     Risposta

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

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

177. Carta 177

     Domanda

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

     Risposta

     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. Carta 178

     Domanda

     Internal rate of return?

     Risposta

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

179. Carta 179

     Domanda

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

     Risposta

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

180. Carta 180

     Domanda

     Convert Celsius temperature to kelvin?

     Risposta

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

     This conversion is for absolute temperature, not temperature difference.

181. Carta 181

     Domanda

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

     Risposta

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

182. Carta 182

     Domanda

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

     Risposta

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

183. Carta 183

     Domanda

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

     Risposta

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

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

184. Carta 184

     Domanda

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

     Risposta

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

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

185. Carta 185

     Domanda

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

     Risposta

     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. Carta 186

     Domanda

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

     Risposta

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

187. Carta 187

     Domanda

     Why inspect boundary or initial conditions near a formula?

     Risposta

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

188. Carta 188

     Domanda

     Convert Celsius temperature to Fahrenheit?

     Risposta

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

189. Carta 189

     Domanda

     Ethical basis for expert technical testimony?

     Risposta

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

190. Carta 190

     Domanda

     Meaning of a breakeven value?

     Risposta

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

191. Carta 191

     Domanda

     Derivative of $e^x$?

     Risposta

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

192. Carta 192

     Domanda

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

     Risposta

     $$\Delta E=Q-W$$

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

193. Carta 193

     Domanda

     Frequentist meaning of a 95% confidence procedure?

     Risposta

     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. Carta 194

     Domanda

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

     Risposta

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

195. Carta 195

     Domanda

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

     Risposta

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

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

196. Carta 196

     Domanda

     Relationship between Celsius and kelvin temperature differences?

     Risposta

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

197. Carta 197

     Domanda

     How should credit for team engineering work be assigned?

     Risposta

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

198. Carta 198

     Domanda

     Main limitation of simple payback period?

     Risposta

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

199. Carta 199

     Domanda

     Roles of the null and alternative hypotheses?

     Risposta

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

200. Carta 200

     Domanda

     Ideal-gas equation of state?

     Risposta

     $$pV=nRT$$

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

201. Carta 201

     Domanda

     First check when reading a handbook table?

     Risposta

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

202. Carta 202

     Domanda

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

     Risposta

     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. Carta 203

     Domanda

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

     Risposta

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

204. Carta 204

     Domanda

     Convert Fahrenheit absolute temperature to Rankine?

     Risposta

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

     Rankine uses the Fahrenheit-sized degree from absolute zero.

205. Carta 205

     Domanda

     Ohm's law for an ideal resistor?

     Risposta

     $$V=IR$$

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

206. Carta 206

     Domanda

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

     Risposta

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

207. Carta 207

     Domanda

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

     Risposta

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

208. Carta 208

     Domanda

     Is a radian dimensionless?

     Risposta

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

209. Carta 209

     Domanda

     Electrical power absorbed under the passive sign convention?

     Risposta

     $$P=VI$$

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

210. Carta 210

     Domanda

     Meaning of a p-value?

     Risposta

     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. Carta 211

     Domanda

     First check when reading a handbook graph?

     Risposta

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

212. Carta 212

     Domanda

     Difference between mass and weight?

     Risposta

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

213. Carta 213

     Domanda

     Kirchhoff's current law at a node?

     Risposta

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

214. Carta 214

     Domanda

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

     Risposta

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

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

215. Carta 215

     Domanda

     Opportunity cost?

     Risposta

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

216. Carta 216

     Domanda

     Weight from mass in a gravitational field?

     Risposta

     $$W=mg$$

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

217. Carta 217

     Domanda

     Kirchhoff's voltage law around a closed loop?

     Risposta

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

218. Carta 218

     Domanda

     Decision rule using significance level $\alpha$?

     Risposta

     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. Carta 219

     Domanda

     Can proprietary methods from a former employer be reused freely?

     Risposta

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

220. Carta 220

     Domanda

     Pound-force to newtons?

     Risposta

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

221. Carta 221

     Domanda

     When does a definite integral equal ordinary geometric area?

     Risposta

     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. Carta 222

     Domanda

     Charge-voltage relation for an ideal capacitor?

     Risposta

     $$q=CV$$

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

223. Carta 223

     Domanda

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

     Risposta

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

224. Carta 224

     Domanda

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

     Risposta

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

225. Carta 225

     Domanda

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

     Risposta

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

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

226. Carta 226

     Domanda

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

     Risposta

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

227. Carta 227

     Domanda

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

     Risposta

     $$\tau=RC$$

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

228. Carta 228

     Domanda

     What physical quantity does pound-mass measure?

     Risposta

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

229. Carta 229

     Domanda

     Type I error?

     Risposta

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

230. Carta 230

     Domanda

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

     Risposta

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

231. Carta 231

     Domanda

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

     Risposta

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

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

232. Carta 232

     Domanda

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

     Risposta

     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. Carta 233

     Domanda

     Recognizable form of a separable first-order differential equation?

     Risposta

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

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

234. Carta 234

     Domanda

     Using unverified software output in a safety-relevant calculation?

     Risposta

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

235. Carta 235

     Domanda

     Equivalent resistance of ideal resistors in series?

     Risposta

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

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

236. Carta 236

     Domanda

     Pressure conversion from psi to pascals?

     Risposta

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

237. Carta 237

     Domanda

     Type II error?

     Risposta

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

238. Carta 238

     Domanda

     Useful flashcard to make after a failed handbook search?

     Risposta

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

239. Carta 239

     Domanda

     Standard form of a first-order linear differential equation?

     Risposta

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

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

240. Carta 240

     Domanda

     Equivalent resistance of ideal resistors in parallel?

     Risposta

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

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

241. Carta 241

     Domanda

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

     Risposta

     $$BV_k=B-kD$$

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

242. Carta 242

     Domanda

     Standard atmosphere in pascals?

     Risposta

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

243. Carta 243

     Domanda

     Using AI-generated technical content in engineering work?

     Risposta

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

244. Carta 244

     Domanda

     Trial form for a homogeneous linear ODE with constant coefficients?

     Risposta

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

245. Carta 245

     Domanda

     Wave-speed relationship among frequency and wavelength?

     Risposta

     $$v=f\lambda$$

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

246. Carta 246

     Domanda

     Statistical power?

     Risposta

     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. Carta 247

     Domanda

     Bar in pascals?

     Risposta

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

248. Carta 248

     Domanda

     How should handbook navigation be practiced before the FE exam?

     Risposta

     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. Carta 249

     Domanda

     Is depreciation itself a cash outflow?

     Risposta

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

250. Carta 250

     Domanda

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

     Risposta

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

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

251. Carta 251

     Domanda

     Relationship between period and frequency?

     Risposta

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

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

252. Carta 252

     Domanda

     How should foreseeable environmental harm enter an engineering decision?

     Risposta

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

253. Carta 253

     Domanda

     Foot-pound-force in joules?

     Risposta

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

254. Carta 254

     Domanda

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

     Risposta

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

255. Carta 255

     Domanda

     When should you search instead of relying on recall?

     Risposta

     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. Carta 256

     Domanda

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

     Risposta

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

257. Carta 257

     Domanda

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

     Risposta

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

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

258. Carta 258

     Domanda

     General material balance on a control volume?

     Risposta

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

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

259. Carta 259

     Domanda

     Approximate joules in one International Table Btu?

     Risposta

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

260. Carta 260

     Domanda

     What is the defender in a replacement analysis?

     Risposta

     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. Carta 261

     Domanda

     Speaking publicly about a technical matter outside your expertise?

     Risposta

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

262. Carta 262

     Domanda

     Single-interval trapezoidal approximation to an integral?

     Risposta

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

     It is exact for linear $f$.

263. Carta 263

     Domanda

     Energy efficiency of a device?

     Risposta

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

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

264. Carta 264

     Domanda

     Purpose of sensitivity analysis in engineering economics?

     Risposta

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

265. Carta 265

     Domanda

     Can a strong correlation establish causation?

     Risposta

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

266. Carta 266

     Domanda

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

     Risposta

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

267. Carta 267

     Domanda

     Mechanical horsepower in watts?

     Risposta

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

268. Carta 268

     Domanda

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

     Risposta

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

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

269. Carta 269

     Domanda

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

     Risposta

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

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

270. Carta 270

     Domanda

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

     Risposta

     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. Carta 271

     Domanda

     Kilowatt-hour in joules?

     Risposta

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

272. Carta 272

     Domanda

     Percent error relative to an accepted nonzero value?

     Risposta

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

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

273. Carta 273

     Domanda

     Does legal compliance alone settle an engineering ethics question?

     Risposta

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

274. Carta 274

     Domanda

     Does handbook fluency replace discipline-specific problem practice?

     Risposta

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

275. Carta 275

     Domanda

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

     Risposta

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

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

276. Carta 276

     Domanda

     Dimensional requirement for a physically meaningful additive equation?

     Risposta

     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.