FE Exam Flashcards: Engineering Equations, Units & Handbook Skills

Kort i denne bunke

1. Kort 1

   Spørgsmål

   SI base unit of length?

   Svar

   The metre, symbol $m$.

2. Kort 2

   Spørgsmål

   Slope between two points?

   Svar

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

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

3. Kort 3

   Spørgsmål

   Net force relationship for constant mass in an inertial frame?

   Svar

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

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

4. Kort 4

   Spørgsmål

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

   Svar

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

5. Kort 5

   Spørgsmål

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

   Svar

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

6. Kort 6

   Spørgsmål

   Probability of the complement of event $A$?

   Svar

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

7. Kort 7

   Spørgsmål

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

   Svar

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

8. Kort 8

   Spørgsmål

   SI base unit of mass?

   Svar

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

9. Kort 9

   Spørgsmål

   Linear equation from slope and intercept?

   Svar

   $$y=mx+b$$

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

10. Kort 10

    Spørgsmål

    Velocity after time $t$ under constant acceleration?

    Svar

    $$v=v_0+at$$

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

11. Kort 11

    Spørgsmål

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

    Svar

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

12. Kort 12

    Spørgsmål

    How is the FE reference handbook available during the exam?

    Svar

    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. Kort 13

    Spørgsmål

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

    Svar

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

14. Kort 14

    Spørgsmål

    SI base unit of time?

    Svar

    The second, symbol $s$.

15. Kort 15

    Spørgsmål

    Linear equation from one point and a slope?

    Svar

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

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

16. Kort 16

    Spørgsmål

    Position after time $t$ under constant acceleration?

    Svar

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

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

17. Kort 17

    Spørgsmål

    A project assignment lies outside your competence. Proper response?

    Svar

    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. Kort 18

    Spørgsmål

    SI base unit of electric current?

    Svar

    The ampere, symbol $A$.

19. Kort 19

    Spørgsmål

    Future amount under simple interest?

    Svar

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

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

20. Kort 20

    Spørgsmål

    Distance between two Cartesian points?

    Svar

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

    The coordinate components must use the same length unit.

21. Kort 21

    Spørgsmål

    Constant-acceleration relation without elapsed time?

    Svar

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

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

22. Kort 22

    Spørgsmål

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

    Svar

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

23. Kort 23

    Spørgsmål

    Conditional probability of $A$ given $B$?

    Svar

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

    This requires $P(B)>0$.

24. Kort 24

    Spørgsmål

    SI base unit of thermodynamic temperature?

    Svar

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

25. Kort 25

    Spørgsmål

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

    Svar

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

26. Kort 26

    Spørgsmål

    Midpoint of two Cartesian points?

    Svar

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

    Average corresponding coordinates.

27. Kort 27

    Spørgsmål

    Linear momentum of a particle?

    Svar

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

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

28. Kort 28

    Spørgsmål

    Future amount under compound interest?

    Svar

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

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

29. Kort 29

    Spørgsmål

    Should an FE search workflow depend on Ctrl+F?

    Svar

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

30. Kort 30

    Spørgsmål

    SI base unit of amount of substance?

    Svar

    The mole, symbol $mol$.

31. Kort 31

    Spørgsmål

    Joint probability expressed with a conditional probability?

    Svar

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

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

32. Kort 32

    Spørgsmål

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

    Svar

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

    The coordinate units and radius unit must match.

33. Kort 33

    Spørgsmål

    Impulse-momentum relationship?

    Svar

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

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

34. Kort 34

    Spørgsmål

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

    Svar

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

35. Kort 35

    Spørgsmål

    SI base unit of luminous intensity?

    Svar

    The candela, symbol $cd$.

36. Kort 36

    Spørgsmål

    Present worth of one future amount?

    Svar

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

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

37. Kort 37

    Spørgsmål

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

    Svar

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

    This form requires $a\ne0$.

38. Kort 38

    Spørgsmål

    When is total linear momentum conserved?

    Svar

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

39. Kort 39

    Spørgsmål

    Why avoid memorizing FE handbook page numbers?

    Svar

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

40. Kort 40

    Spørgsmål

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

    Svar

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

    Independence is not the same as mutual exclusivity.

41. Kort 41

    Spørgsmål

    SI unit of frequency in base-unit form?

    Svar

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

42. Kort 42

    Spørgsmål

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

    Svar

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

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

43. Kort 43

    Spørgsmål

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

    Svar

    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. Kort 44

    Spørgsmål

    Work done by a constant force through displacement?

    Svar

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

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

45. Kort 45

    Spørgsmål

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

    Svar

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

46. Kort 46

    Spørgsmål

    SI unit of force in base-unit form?

    Svar

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

47. Kort 47

    Spørgsmål

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

    Svar

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

48. Kort 48

    Spørgsmål

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

    Svar

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

    This requires $P(B)>0$.

49. Kort 49

    Spørgsmål

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

    Svar

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

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

50. Kort 50

    Spørgsmål

    Translational kinetic energy of a particle?

    Svar

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

    Mass and speed must use a coherent unit system.

51. Kort 51

    Spørgsmål

    SI unit of pressure in force-area form?

    Svar

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

52. Kort 52

    Spørgsmål

    Equivalent annual amount for present worth $P$?

    Svar

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

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

53. Kort 53

    Spørgsmål

    May confidential client information be shared for personal benefit?

    Svar

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

54. Kort 54

    Spørgsmål

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

    Svar

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

55. Kort 55

    Spørgsmål

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

    Svar

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

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

56. Kort 56

    Spørgsmål

    Net work-energy relationship for a particle?

    Svar

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

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

57. Kort 57

    Spørgsmål

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

    Svar

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

    Use this when order matters and selection is without replacement.

58. Kort 58

    Spørgsmål

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

    Svar

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

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

59. Kort 59

    Spørgsmål

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

    Svar

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

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

60. Kort 60

    Spørgsmål

    SI unit of energy in force-distance form?

    Svar

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

61. Kort 61

    Spørgsmål

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

    Svar

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

62. Kort 62

    Spørgsmål

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

    Svar

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

63. Kort 63

    Spørgsmål

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

    Svar

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

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

64. Kort 64

    Spørgsmål

    Change in gravitational potential energy near Earth's surface?

    Svar

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

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

65. Kort 65

    Spørgsmål

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

    Svar

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

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

66. Kort 66

    Spørgsmål

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

    Svar

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

    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. Kort 67

    Spørgsmål

    A handbook search for an acronym fails. Next move?

    Svar

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

68. Kort 68

    Spørgsmål

    SI unit of power in energy-time form?

    Svar

    The watt, $W=J/s$.

69. Kort 69

    Spørgsmål

    Logarithm of a product?

    Svar

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

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

70. Kort 70

    Spørgsmål

    Conditions for a binomial model?

    Svar

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

71. Kort 71

    Spørgsmål

    Using another engineer's analysis in your deliverable?

    Svar

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

72. Kort 72

    Spørgsmål

    SI unit of electric charge in current-time form?

    Svar

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

73. Kort 73

    Spørgsmål

    Average power from work over elapsed time?

    Svar

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

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

74. Kort 74

    Spørgsmål

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

    Svar

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

    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. Kort 75

    Spørgsmål

    Logarithm of a positive quantity raised to a power?

    Svar

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

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

76. Kort 76

    Spørgsmål

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

    Svar

    The volt, $V=J/C$.

77. Kort 77

    Spørgsmål

    Instantaneous mechanical power delivered by a force?

    Svar

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

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

78. Kort 78

    Spørgsmål

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

    Svar

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

79. Kort 79

    Spørgsmål

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

    Svar

    $$E[X]=np$$

80. Kort 80

    Spørgsmål

    SI unit of electrical resistance in voltage-current form?

    Svar

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

81. Kort 81

    Spørgsmål

    Moment of a force about a point?

    Svar

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

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

82. Kort 82

    Spørgsmål

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

    Svar

    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. Kort 83

    Spørgsmål

    Inverse relationship between the natural log and exponential?

    Svar

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

    The second identity requires $x>0$.

84. Kort 84

    Spørgsmål

    SI prefix giga?

    Svar

    $G=10^9$.

85. Kort 85

    Spørgsmål

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

    Svar

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

    Use consistent angular sign convention and radians for angular measures.

86. Kort 86

    Spørgsmål

    When should an engineer sign or seal technical work?

    Svar

    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. Kort 87

    Spørgsmål

    Best memory target for returning to a handbook formula?

    Svar

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

88. Kort 88

    Spørgsmål

    SI prefix mega?

    Svar

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

89. Kort 89

    Spørgsmål

    Convert degrees to radians?

    Svar

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

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

90. Kort 90

    Spørgsmål

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

    Svar

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

91. Kort 91

    Spørgsmål

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

    Svar

    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. Kort 92

    Spørgsmål

    Where do licensure and sealing requirements come from?

    Svar

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

93. Kort 93

    Spørgsmål

    Rotational kinetic energy about a fixed axis?

    Svar

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

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

94. Kort 94

    Spørgsmål

    SI prefix kilo?

    Svar

    $k=10^3$.

95. Kort 95

    Spørgsmål

    Right-triangle definition of sine?

    Svar

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

96. Kort 96

    Spørgsmål

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

    Svar

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

97. Kort 97

    Spørgsmål

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

    Svar

    $$\sum M=I\alpha$$

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

98. Kort 98

    Spørgsmål

    SI prefix milli?

    Svar

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

99. Kort 99

    Spørgsmål

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

    Svar

    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. Kort 100

     Spørgsmål

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

     Svar

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

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

101. Kort 101

     Spørgsmål

     Right-triangle definition of cosine?

     Svar

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

102. Kort 102

     Spørgsmål

     Force-displacement relation for an ideal linear spring?

     Svar

     $$F=-kx$$

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

103. Kort 103

     Spørgsmål

     May qualifications or project responsibility be exaggerated in a proposal?

     Svar

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

104. Kort 104

     Spørgsmål

     SI prefix micro?

     Svar

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

105. Kort 105

     Spørgsmål

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

     Svar

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

106. Kort 106

     Spørgsmål

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

     Svar

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

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

107. Kort 107

     Spørgsmål

     Right-triangle definition of tangent?

     Svar

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

108. Kort 108

     Spørgsmål

     Average normal stress under centered axial load?

     Svar

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

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

109. Kort 109

     Spørgsmål

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

     Svar

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

110. Kort 110

     Spørgsmål

     SI prefix nano?

     Svar

     $n=10^{-9}$.

111. Kort 111

     Spørgsmål

     Fundamental Pythagorean trigonometric identity?

     Svar

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

112. Kort 112

     Spørgsmål

     Engineering normal strain?

     Svar

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

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

113. Kort 113

     Spørgsmål

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

     Svar

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

114. Kort 114

     Spørgsmål

     Technical duty to an employer or client?

     Svar

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

115. Kort 115

     Spørgsmål

     How does a length conversion factor apply to area?

     Svar

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

116. Kort 116

     Spørgsmål

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

     Svar

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

117. Kort 117

     Spørgsmål

     When is a Poisson count model reasonable?

     Svar

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

118. Kort 118

     Spørgsmål

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

     Svar

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

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

119. Kort 119

     Spørgsmål

     Linear elastic stress-strain relation in uniaxial loading?

     Svar

     $$\sigma=E\varepsilon$$

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

120. Kort 120

     Spørgsmål

     How does a length conversion factor apply to volume?

     Svar

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

121. Kort 121

     Spørgsmål

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

     Svar

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

122. Kort 122

     Spørgsmål

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

     Svar

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

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

123. Kort 123

     Spørgsmål

     A supervisor directs an illegal engineering action. Proper response?

     Svar

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

124. Kort 124

     Spørgsmål

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

     Svar

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

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

125. Kort 125

     Spørgsmål

     Mass density?

     Svar

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

     Use mass, not weight, in the numerator.

126. Kort 126

     Spørgsmål

     Inches in one foot?

     Svar

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

127. Kort 127

     Spørgsmål

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

     Svar

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

128. Kort 128

     Spørgsmål

     How should subscripts be handled in a retrieved relationship?

     Svar

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

129. Kort 129

     Spørgsmål

     Dot product from vector magnitudes and included angle?

     Svar

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

     The result is a scalar.

130. Kort 130

     Spørgsmål

     Average normal pressure on an area?

     Svar

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

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

131. Kort 131

     Spørgsmål

     Feet in one yard?

     Svar

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

132. Kort 132

     Spørgsmål

     When are two interest rates equivalent?

     Svar

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

133. Kort 133

     Spørgsmål

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

     Svar

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

134. Kort 134

     Spørgsmål

     Expected value of a discrete random variable?

     Svar

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

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

135. Kort 135

     Spørgsmål

     Dot-product test for perpendicular nonzero vectors?

     Svar

     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. Kort 136

     Spørgsmål

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

     Svar

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

     Depth increases downward and the fluid is at rest.

137. Kort 137

     Spørgsmål

     Feet in one statute mile?

     Svar

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

138. Kort 138

     Spørgsmål

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

     Svar

     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. Kort 139

     Spørgsmål

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

     Svar

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

140. Kort 140

     Spørgsmål

     Magnitude of a vector cross product?

     Svar

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

     The direction follows the right-hand rule.

141. Kort 141

     Spørgsmål

     Exact inch-to-metre relationship?

     Svar

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

142. Kort 142

     Spørgsmål

     Volumetric flow rate through a cross-section?

     Svar

     $$Q=Av_{avg}$$

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

143. Kort 143

     Spørgsmål

     Variance from the mean $\mu$?

     Svar

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

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

144. Kort 144

     Spørgsmål

     Future-worth acceptance rule for an independent investment?

     Svar

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

145. Kort 145

     Spørgsmål

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

     Svar

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

146. Kort 146

     Spørgsmål

     Exact foot-to-metre relationship?

     Svar

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

147. Kort 147

     Spørgsmål

     How should important uncertainty be communicated?

     Svar

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

148. Kort 148

     Spørgsmål

     When is the matrix product $AB$ defined?

     Svar

     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. Kort 149

     Spørgsmål

     Steady one-inlet, one-outlet mass continuity?

     Svar

     $$\rho_1A_1v_1=\rho_2A_2v_2$$

     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. Kort 150

     Spørgsmål

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

     Svar

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

151. Kort 151

     Spørgsmål

     Square feet in one acre?

     Svar

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

152. Kort 152

     Spørgsmål

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

     Svar

     $$ad-bc$$

153. Kort 153

     Spørgsmål

     Bernoulli relationship along a streamline for ideal steady flow?

     Svar

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

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

154. Kort 154

     Spørgsmål

     May inconvenient measurements be discarded without a technical basis?

     Svar

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

155. Kort 155

     Spørgsmål

     Annual-worth acceptance rule for an independent investment?

     Svar

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

156. Kort 156

     Spørgsmål

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

     Svar

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

157. Kort 157

     Spørgsmål

     Cubic metres in one litre?

     Svar

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

158. Kort 158

     Spørgsmål

     What does a zero determinant mean for a square matrix?

     Svar

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

159. Kort 159

     Spørgsmål

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

     Svar

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

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

160. Kort 160

     Spørgsmål

     Unbiased sample variance for an independent sample from one population?

     Svar

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

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

161. Kort 161

     Spørgsmål

     How should mutually exclusive alternatives be compared?

     Svar

     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. Kort 162

     Spørgsmål

     Cubic centimetres in one millilitre?

     Svar

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

163. Kort 163

     Spørgsmål

     Why keep contemporaneous engineering records?

     Svar

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

164. Kort 164

     Spørgsmål

     Fast check after substituting into a handbook formula?

     Svar

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

165. Kort 165

     Spørgsmål

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

     Svar

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

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

166. Kort 166

     Spørgsmål

     Sensible heat for constant specific heat and no phase change?

     Svar

     $$Q=mc\Delta T$$

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

167. Kort 167

     Spørgsmål

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

     Svar

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

     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. Kort 168

     Spørgsmål

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

     Svar

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

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

169. Kort 169

     Spørgsmål

     Purpose of incremental analysis for mutually exclusive alternatives?

     Svar

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

170. Kort 170

     Spørgsmål

     Physical meaning of $dy/dx$?

     Svar

     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. Kort 171

     Spørgsmål

     One-dimensional steady conduction through a plane wall?

     Svar

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

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

172. Kort 172

     Spørgsmål

     When should you algebraically rearrange a handbook relationship?

     Svar

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

173. Kort 173

     Spørgsmål

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

     Svar

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

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

174. Kort 174

     Spørgsmål

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

     Svar

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

175. Kort 175

     Spørgsmål

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

     Svar

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

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

176. Kort 176

     Spørgsmål

     Convective heat-transfer rate at a surface?

     Svar

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

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

177. Kort 177

     Spørgsmål

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

     Svar

     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. Kort 178

     Spørgsmål

     Internal rate of return?

     Svar

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

179. Kort 179

     Spørgsmål

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

     Svar

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

180. Kort 180

     Spørgsmål

     Convert Celsius temperature to kelvin?

     Svar

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

     This conversion is for absolute temperature, not temperature difference.

181. Kort 181

     Spørgsmål

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

     Svar

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

182. Kort 182

     Spørgsmål

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

     Svar

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

183. Kort 183

     Spørgsmål

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

     Svar

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

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

184. Kort 184

     Spørgsmål

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

     Svar

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

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

185. Kort 185

     Spørgsmål

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

     Svar

     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. Kort 186

     Spørgsmål

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

     Svar

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

187. Kort 187

     Spørgsmål

     Why inspect boundary or initial conditions near a formula?

     Svar

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

188. Kort 188

     Spørgsmål

     Convert Celsius temperature to Fahrenheit?

     Svar

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

189. Kort 189

     Spørgsmål

     Ethical basis for expert technical testimony?

     Svar

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

190. Kort 190

     Spørgsmål

     Meaning of a breakeven value?

     Svar

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

191. Kort 191

     Spørgsmål

     Derivative of $e^x$?

     Svar

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

192. Kort 192

     Spørgsmål

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

     Svar

     $$\Delta E=Q-W$$

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

193. Kort 193

     Spørgsmål

     Frequentist meaning of a 95% confidence procedure?

     Svar

     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. Kort 194

     Spørgsmål

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

     Svar

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

195. Kort 195

     Spørgsmål

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

     Svar

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

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

196. Kort 196

     Spørgsmål

     Relationship between Celsius and kelvin temperature differences?

     Svar

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

197. Kort 197

     Spørgsmål

     How should credit for team engineering work be assigned?

     Svar

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

198. Kort 198

     Spørgsmål

     Main limitation of simple payback period?

     Svar

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

199. Kort 199

     Spørgsmål

     Roles of the null and alternative hypotheses?

     Svar

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

200. Kort 200

     Spørgsmål

     Ideal-gas equation of state?

     Svar

     $$pV=nRT$$

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

201. Kort 201

     Spørgsmål

     First check when reading a handbook table?

     Svar

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

202. Kort 202

     Spørgsmål

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

     Svar

     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. Kort 203

     Spørgsmål

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

     Svar

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

204. Kort 204

     Spørgsmål

     Convert Fahrenheit absolute temperature to Rankine?

     Svar

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

     Rankine uses the Fahrenheit-sized degree from absolute zero.

205. Kort 205

     Spørgsmål

     Ohm's law for an ideal resistor?

     Svar

     $$V=IR$$

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

206. Kort 206

     Spørgsmål

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

     Svar

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

207. Kort 207

     Spørgsmål

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

     Svar

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

208. Kort 208

     Spørgsmål

     Is a radian dimensionless?

     Svar

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

209. Kort 209

     Spørgsmål

     Electrical power absorbed under the passive sign convention?

     Svar

     $$P=VI$$

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

210. Kort 210

     Spørgsmål

     Meaning of a p-value?

     Svar

     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. Kort 211

     Spørgsmål

     First check when reading a handbook graph?

     Svar

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

212. Kort 212

     Spørgsmål

     Difference between mass and weight?

     Svar

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

213. Kort 213

     Spørgsmål

     Kirchhoff's current law at a node?

     Svar

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

214. Kort 214

     Spørgsmål

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

     Svar

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

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

215. Kort 215

     Spørgsmål

     Opportunity cost?

     Svar

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

216. Kort 216

     Spørgsmål

     Weight from mass in a gravitational field?

     Svar

     $$W=mg$$

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

217. Kort 217

     Spørgsmål

     Kirchhoff's voltage law around a closed loop?

     Svar

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

218. Kort 218

     Spørgsmål

     Decision rule using significance level $\alpha$?

     Svar

     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. Kort 219

     Spørgsmål

     Can proprietary methods from a former employer be reused freely?

     Svar

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

220. Kort 220

     Spørgsmål

     Pound-force to newtons?

     Svar

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

221. Kort 221

     Spørgsmål

     When does a definite integral equal ordinary geometric area?

     Svar

     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. Kort 222

     Spørgsmål

     Charge-voltage relation for an ideal capacitor?

     Svar

     $$q=CV$$

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

223. Kort 223

     Spørgsmål

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

     Svar

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

224. Kort 224

     Spørgsmål

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

     Svar

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

225. Kort 225

     Spørgsmål

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

     Svar

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

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

226. Kort 226

     Spørgsmål

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

     Svar

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

227. Kort 227

     Spørgsmål

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

     Svar

     $$\tau=RC$$

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

228. Kort 228

     Spørgsmål

     What physical quantity does pound-mass measure?

     Svar

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

229. Kort 229

     Spørgsmål

     Type I error?

     Svar

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

230. Kort 230

     Spørgsmål

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

     Svar

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

231. Kort 231

     Spørgsmål

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

     Svar

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

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

232. Kort 232

     Spørgsmål

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

     Svar

     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. Kort 233

     Spørgsmål

     Recognizable form of a separable first-order differential equation?

     Svar

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

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

234. Kort 234

     Spørgsmål

     Using unverified software output in a safety-relevant calculation?

     Svar

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

235. Kort 235

     Spørgsmål

     Equivalent resistance of ideal resistors in series?

     Svar

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

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

236. Kort 236

     Spørgsmål

     Pressure conversion from psi to pascals?

     Svar

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

237. Kort 237

     Spørgsmål

     Type II error?

     Svar

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

238. Kort 238

     Spørgsmål

     Useful flashcard to make after a failed handbook search?

     Svar

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

239. Kort 239

     Spørgsmål

     Standard form of a first-order linear differential equation?

     Svar

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

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

240. Kort 240

     Spørgsmål

     Equivalent resistance of ideal resistors in parallel?

     Svar

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

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

241. Kort 241

     Spørgsmål

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

     Svar

     $$BV_k=B-kD$$

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

242. Kort 242

     Spørgsmål

     Standard atmosphere in pascals?

     Svar

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

243. Kort 243

     Spørgsmål

     Using AI-generated technical content in engineering work?

     Svar

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

244. Kort 244

     Spørgsmål

     Trial form for a homogeneous linear ODE with constant coefficients?

     Svar

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

245. Kort 245

     Spørgsmål

     Wave-speed relationship among frequency and wavelength?

     Svar

     $$v=f\lambda$$

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

246. Kort 246

     Spørgsmål

     Statistical power?

     Svar

     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. Kort 247

     Spørgsmål

     Bar in pascals?

     Svar

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

248. Kort 248

     Spørgsmål

     How should handbook navigation be practiced before the FE exam?

     Svar

     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. Kort 249

     Spørgsmål

     Is depreciation itself a cash outflow?

     Svar

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

250. Kort 250

     Spørgsmål

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

     Svar

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

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

251. Kort 251

     Spørgsmål

     Relationship between period and frequency?

     Svar

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

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

252. Kort 252

     Spørgsmål

     How should foreseeable environmental harm enter an engineering decision?

     Svar

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

253. Kort 253

     Spørgsmål

     Foot-pound-force in joules?

     Svar

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

254. Kort 254

     Spørgsmål

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

     Svar

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

255. Kort 255

     Spørgsmål

     When should you search instead of relying on recall?

     Svar

     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. Kort 256

     Spørgsmål

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

     Svar

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

257. Kort 257

     Spørgsmål

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

     Svar

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

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

258. Kort 258

     Spørgsmål

     General material balance on a control volume?

     Svar

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

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

259. Kort 259

     Spørgsmål

     Approximate joules in one International Table Btu?

     Svar

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

260. Kort 260

     Spørgsmål

     What is the defender in a replacement analysis?

     Svar

     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. Kort 261

     Spørgsmål

     Speaking publicly about a technical matter outside your expertise?

     Svar

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

262. Kort 262

     Spørgsmål

     Single-interval trapezoidal approximation to an integral?

     Svar

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

     It is exact for linear $f$.

263. Kort 263

     Spørgsmål

     Energy efficiency of a device?

     Svar

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

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

264. Kort 264

     Spørgsmål

     Purpose of sensitivity analysis in engineering economics?

     Svar

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

265. Kort 265

     Spørgsmål

     Can a strong correlation establish causation?

     Svar

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

266. Kort 266

     Spørgsmål

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

     Svar

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

267. Kort 267

     Spørgsmål

     Mechanical horsepower in watts?

     Svar

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

268. Kort 268

     Spørgsmål

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

     Svar

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

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

269. Kort 269

     Spørgsmål

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

     Svar

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

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

270. Kort 270

     Spørgsmål

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

     Svar

     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. Kort 271

     Spørgsmål

     Kilowatt-hour in joules?

     Svar

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

272. Kort 272

     Spørgsmål

     Percent error relative to an accepted nonzero value?

     Svar

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

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

273. Kort 273

     Spørgsmål

     Does legal compliance alone settle an engineering ethics question?

     Svar

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

274. Kort 274

     Spørgsmål

     Does handbook fluency replace discipline-specific problem practice?

     Svar

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

275. Kort 275

     Spørgsmål

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

     Svar

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

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

276. Kort 276

     Spørgsmål

     Dimensional requirement for a physically meaningful additive equation?

     Svar

     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.