Updated for the new July 2020 FE Exam Topics

Free FE Practice Test

# PrepFE™

## Free FE Other Disciplines Example Practice Problems

We've selected 10 diverse practice problems from our question bank that you can use to review for the Other Disciplines engineering FE exam and give you an idea about some of the content we provide.

## Solutions

### 1) Balance this chemical reaction by choosing the right variables for the blanks $\rule{0.5cm}{0.2mm} CO_2 + \rule{0.5cm}{0.2mm} C \rightarrow \space \rule{0.5cm}{0.2mm} CO$.

A.1 1 1
B.1 1 2
C.1 2 1
D.1 2 2
The correct answer is B.

### Explanation:

This is a simpler stoichiometry problem and is best solved quickly with a visual inspection, and that skill will come in time. Let's break it down anyway, as this is a useful skill for more complex problems. Essentially you are setting up and then solving a system of equations. If we look at just the carbon (C) atoms, then we have the equation: $$xC + yC = zC \Rightarrow x + y = 2z$$ Lots of variables for x, y, and z could solve this equation. Let's look at the Oxygen (O) atoms now and see if it's helpful: $$a(O_2) = b(O)$$ We can see that the relationship between a and b is: $$a = 2b$$ We also know that b = z and that a = x. Let's substitute what we know into the first equation: $$a + y = 2b$$ And since x = a, we can solve for y: $$b + y = 2b \\ \Rightarrow y = b$$ Now use only b throughout the first equation. $$b + b = 2b$$ So other than the trivial answer of zero, $b=1$ is the first integer that solves it and is the best solution. We finally have: $$CO_2 + C \rightarrow 2CO$$
Note that you could have solved for other variables besides b and still gotten to the same final solution.

### 2) Given the values below, what is nearest the total power dissipated for this circuit? $R_1 = \SI{10}{\Omega}$, $R_2 = \SI{3}{\Omega}$, $R_3 = \SI{8}{\Omega}$ $V_1 = \SI{5}{V}$, $V_2 = \SI{8}{V}$

A.1.1 W
B.3.0 W
C.7.1 W
D.21 W
The correct answer is B.

### Explanation:

$$P = \frac{V^2}{R}$$ $$= \frac{(V_2-V_1)^2}{R_3} + \frac{(V_1)^2}{R_1+R_2}$$ $$= \frac{(\SI{8}{V}-\SI{5}{V})^2}{\SI{8}{\Omega}} + \frac{(\SI{5}{V})^2}{\SI{13}{\Omega}}$$ $$\approx \SI{3.05}{W}$$

### 3) A total of 50 valve pressure measurements were taken but only 9 of those measurements will be utilized. What is most nearly the standard deviation of the sample measurements? $$\begin{array}{c|c} \text{Pressure} & \text{# of Measurements} \\ \hline 20 & 3 \\ 22 & 4 \\ 25 & 2 \\ \end{array}$$

A.0.51
B.2.18
C.1.93
D.1.20
The correct answer is C.

### Explanation:

Refer to the Dispersion, Mean, Median, and Mode Values section in the Engineering Probability and Statistics chapter of the FE Reference Handbook.

The population is 50, but the sample is 9. Since we only have pressure values for 9 measurements out of the 50 total measurements, we must use the sample standard deviation formula. Ultimately, we must solve for: $$s = \sqrt{ \frac{1}{(n-1)} \sum_{i=1}^n \left(X_i-\bar{X}\right)^2 }$$ Solve for the unknowns.
The total number of samples is $n=9$.
Calculate the arithmetic mean, $\bar{X}.$ $$\bar{X}=(1/n)(X_1+X_2+...+X_n) \\ \bar{X}=(1/9)(20+20+20+22+ \\ 22+22+22+25+25)$$ $$\bar{X} = 22$$ Now that we have all the unknowns, solve for standard deviation. $$s = \sqrt{ \frac{1}{9-1} \left[ 3(20-22)^2 + 4(22-22)^2 + 2(25-22)^2 \right] }$$ $$\approx 1.93$$
Note: You can use your TI-36X pro calculator's standard deviation function to solve this problem quicker.

### 4) The following three lines form a triangle with vertices at what points? $$x+3y=1$$ $$2x+y=0$$ $$-2x+y=0$$

A.$(- \frac{1}{5},\frac{2}{5}), (0,0), (\frac{1}{7}, \frac{2}{7})$
B.$(- \frac{1}{5},\frac{2}{5}), (1,1), (\frac{1}{7}, \frac{2}{7})$
C.$(- \frac{1}{5},-\frac{2}{5}), (0,1), (\frac{1}{7}, \frac{2}{7})$
D.$(\frac{1}{5},\frac{2}{5}), (0,0), (\frac{1}{7}, \frac{2}{7})$
The correct answer is A.

### Explanation:

Find the intersection for each pair of lines. Using the initial two equations, let's try substituting the second into the first by solving the second for y: $$y = -2x \\ \Rightarrow x+3(-2x)=1 \\ -5x = 1 \\ x = - \frac{1}{5}$$ Then plug that back into either of the first two equations to find the y-coordinate of the intersection. $$y = -2(- \frac{1}{5}) = \frac{2}{5} \\ \rightarrow (x,y)=(- \frac{1}{5},\frac{2}{5})$$ Now we do the same thing twice for the second and third equations and then the first and third equations. For the second and third equations, we see that $y=-y$, so the second intersection we know that only (0,0) will work. For the last pair: $$y = 2x \\ \Rightarrow x+3(2x)=1 \\ 7x = 1 \\ x = \frac{1}{7} \\ \rightarrow (x,y) = (\frac{1}{7}, \frac{2}{7})$$ We have: $$(- \frac{1}{5},\frac{2}{5}), (0,0), (\frac{1}{7}, \frac{2}{7})$$

### 5) An engineering firm plans to submit bids for two different contracts. The cost of preparing both of the bids is \$12,000 total. Each contract is worth \$100,000. If the chances of winning each bid is 10%. What is the expected value for the money earned by the firm? $$\begin{array}{c|c|c} \text{Bids Won} &0&1&2\\ \hline \text{\ Earned} & -\12,000 & \88,000 & \188,000 \\P(x) & 0.81 & 0.18 &0.01 \\ \end{array}$$

A.\$8,000 B.\$22,000
C.\$12,000 D.\$0
The correct answer is A.

### Explanation:

Expected value is simply calculated by multiplying each of the possible outcomes by the likelihood each outcome will occur and then summing all of those values. $$E(x) = \sum F_i P_i\\ = F_1 P_1+ F_2 P_2 + F_3P_3 \\ = -\\text{12,000}(0.81) + \\text{88,000}(0.18) + \\text{188,000}(0.01) \\ = \\text{8,000}$$ In this problem, $P_i$ is provided, but you could calculate this from the problem statement if it wasn't provided in the table. This is because it states that each bid has a 10% chance of winning.

First, what are the chances that we lose both bids ($x=0$). If winning has a 10% chance, then losing has a 90% chance. $$P(0) = 0.9 \cdot 0.9 = 0.81$$ What about when one bid loses and one bid wins? $$P = 0.9 \cdot 0.1 = 0.09$$ Careful! We must also include the reverse scenario: one bid wins and one bid loses, $P = 0.1 \cdot 0.9 = 0.09$. Combined we now have P(1): $$P(1) = 0.9 \cdot 0.1 + 0.1 \cdot 0.9 = 0.18$$ And lastly, what are the chances if we win both bids? $$P(2) = 0.1 \cdot 0.1 = 0.01$$

Once you work through the example, you see it's easier to understand than what the formal definition of expected value presented in FE Reference Handbook makes it seem.

The expected value is also known as the expectation, mathematical expectation, mean, or first moment. It is the probability-weighted average of a discrete random variable's possible values.

### 6) Flow in a circular pipe of diameter $\SI{8}{cm}$ has an average velocity of $\SI{2.5}{cm/s}$. What is the velocity through this pipe if the diameter was 1/4th the current size? at $r=D/4$? Assume laminar flow conditions.

A.2.5 cm/s
B.3.7 cm/s
C.4.4 cm/s
D.5.0 cm/s
The correct answer is B.

### Explanation:

Refer to Fluid Flow Characterization section in the Fluid Mechanics chapter of the FE Reference Handbook.

Laminar flow in circular tubes has velocity distribution that follows: $$v(r) = v_{max}\left[1-\left(\frac{r}{R}\right)^2\right]$$ where
$v_{max} = 2\bar{v}=2(2.5\si{cm/s})=5\si{cm/s}$
Current tube diameter = $8\si{cm}$
Current tube radius = $8\si{cm}/2=4\si{cm}$
$R=\text{radius of new tube}=(1/4)(4\si{cm})=1\si{cm}$
$r=\text{distance from the centerline}=(1/4)(4\si{cm})=1\si{cm}$ $$v\left(\SI{2}{cm}\right) = 2\bar{v}\left[1-\left(\frac{\frac{D}{4}}{\frac{D}{2}}\right)^2\right]$$ $$v\left(\SI{2}{cm}\right) = 2(\SI{2.5}{cm/s})\left[1-\left(\frac{\frac{8\si{cm}}{4}}{\frac{8\si{cm}}{2}}\right)^2\right]$$ $$\approx \SI{3.75}{cm/s}$$

Side note: Using the continuity equation $v_1a_1=v_2a_2$ to solve this problem would've been incorrect as the continuity equation is used for calculating flows between two points . This problem only has one point, and we just want to know what velocity would be like at the same point given a different pipe diameter.

### 7) What is approximately the velocity of the incompressible flow at $x_2$ if the flow exits the pipe at atmospheric pressure and $h = \SI{0.5}{m}$ ignoring friction?

A.3.1 m/s
B.6.1 m/s
C.7.0 m/s
D.11 m/s
The correct answer is A.

### Explanation:

Refer to the Principles of One-Dimensional Fluid Flow section in the Fluid Mechanics chapter of the FE Reference Handbook.

Ultimately, we must solve for the velocity at the outlet of the system, $v_2$. Use Bernoulli's equation to solve for $v_2.$ $$\frac{p_2}{\gamma} +\frac{v_2^2} {2g} +z_2 = \frac{p_1}{\gamma} + \frac{v_1^2} {2g} +z_1$$ Calculate the unknowns:

$p_1=0 \si{Pa}$ because the water is exposed to the atmosphere, which is 0 gauge pressure or 101.3 kPa absolute pressure
$\gamma_{water}=9.81\si{kN/m^3}$
$z_1=0.5\si{m}$
$g=9.81\si{m/s^2}$
$v_1=0$ since the water at $x_1$ is just sitting there, not moving.
$v_2=?$
$p_2=0 \si{Pa}$ because the water is released to the atmosphere, which is 0 gauge pressure or 101.3 kPa absolute pressure
$z_2=0$ $$\frac{0}{9.81\si{kN/m^3}} +\frac{v_2^2} {2(9.81\si{m/s^2})} +0 = \frac{0}{9.81\si{kN/m^3}} \\ + \frac{0^2} {2(9.81\si{m/s^2})} +0.5\si{m}; \text{solve for }v_2$$ $$v_2=3.1\si{m/s}$$

### 8) In the stress-strain diagram, select all that point A describes.

The correct answers are A and B.

### Explanation:

Along A, a material's behavior is elastic - its deformation is not permanent. The slope of A is used to find Young's modulus.

### 9) An aluminum rod with diameter of $d = \SI{12}{\milli\meter}$ is loaded axially in compression as shown. Using a Young's modulus of $\SI{70}{\giga\pascal}$ and Poisson ratio of 0.35, what is most nearly the increase in diameter of the rod at the midspan of the rod? Assume the internal loading at the midspan of the rod is -500 kN.

A.$\SI{49}{\micro\meter}$
B.0.31 mm
C.$\SI{265}{\micro\meter}$
D.230 nm
The correct answer is C.

### Explanation:

Refer to the Uniaxial Stress-Strain section in the Mechanics of Materials chapter of the FE Reference Handbook.

Ultimately, we must determine the increase in rod diameter after the rod is compressed. We can determine the increase in length of a member from its engineering strain. $$\epsilon=\Delta L/L_0$$ Which can be interpreted as $$\epsilon_{lateral}=\Delta L_{lateral}/L_{0 lateral}=\Delta d/d_0$$ Solve for the unknowns. Ultimately, we must solve for $\Delta d$.
$d_0=12\si{mm}=0.012\si{m} \\ \Delta d=? \\ \epsilon_{lateral}=?$

Side note: lateral strain refers to strain that occurs to the sides of the rod (cross section) and NOT strain that occurs along the length of the rod, that would be longitudinal strain. Therefore the equation above was set up in terms of lateral strain because this problem asks for the change in length laterally (at the cross section). Solve for lateral strain using Poisson's ratio. $$\nu=\frac{-\text{lateral strain}}{\text{longitudinal strain}}=0.35=\frac{-\text{lateral strain}}{\text{longitudinal strain}}$$ Solve for longitudinal strain using Hooke's law, so we can then come back and solve for lateral strain. $$E=\sigma/ \epsilon_{longitudinal}\rightarrow E=(P/A)/\epsilon_{longitudinal}$$ Solve for the unknowns.
$E= 70\times 10^9\si{Pa} \\ A=\pi/4\cdot d^2=\pi/4\cdot (0.012\si{m})^2=0.000113097 \si{m^2} \\ P=\text{internal loading}=500\si{kN}=500,000\si{N}$ $$70\times 10^9\si{Pa}=(-500,000\si{N}/0.000113097 \si{m^2})/\epsilon_{longitudinal} \\ \therefore \epsilon_{longitudinal}=-0.0631569$$ Now we can go back and solve for lateral strain using Poisson's ratio. $$0.35=\frac{-\text{lateral strain}}{\text{longitudinal strain}}=\frac{-\epsilon_{lateral}}{-0.0631569}$$ $$\therefore \epsilon_{lateral}=0.022104915$$ Now we can go back to the first equation and solve for the change in diameter. $$\epsilon_{lateral}=\Delta d/d_0 \\ 0.022104915=\Delta d/0.012\si{m};\text{solve for } \Delta d \\ \Delta d = 0.000265259\si{m}=\SI{265}{\micro\meter}$$ Some of the takeaways from this problem are:
Strain and deformation to a rod occur longitudinally (along the length of the rod) or laterally (along the cross section diameter length). Therefore, you can use any of the Hooke's Law equations and modify them to be in terms of lateral deformation or longitudinal deformation. In this problem, the rod got shorter and the diameter got larger as the two 500 kN pushed into the rod axially.

### 10) A parallel flow heat exchanger operates with water ($c_{p,w} = \SI{4.186}{\kilo\joule\per\kilogram\per\kelvin}$) and oil ($c_{p,o} = \SI{3.4}{\kilo\joule\per\kilogram\per\kelvin}$). Water, the cold fluid, enters the inner tube at $\SI{300}{\kelvin}$ and leaves at $\SI{400}{\kelvin}$. What is the log mean temperature difference if oil, the hot fluid, enters the heat exchanger at $\SI{550}{\kelvin}$ and leaves the system at $\SI{450}{\kelvin}$?

A.$\SI{240}{\kelvin}$
B.$\SI{234}{\kelvin}$
C.$\SI{108}{\kelvin}$
D.$\SI{124}{\kelvin}$
The correct answer is D.

### Explanation:

According to the FE Reference Handbook, the log mean temperature difference for parallel flow heat exchangers is defined as $$\Delta T_{lm} = {(T_{Ho} - T_{Co}) - (T_{Hi} - T_{Ci}) \over ln \left ( T_{Ho} - T_{Co} \over T_{Hi} - T_{Ci} \right )}$$ where $T_{Hi}$ the hot fluid inlet temperature, $T_{Ho}$ the hot fluid outlet temperature, $T_{Ci}$ the cold fluid inlet temperature and $T_{Co}$ the cold fluid outlet temperature. In our problem water is the cold fluid and oil the hot one, therefore we have $$\Delta T_{lm} = {(T_{oil,out} - T_{water,out}) - (T_{oil,in} - T_{water,in}) \over ln \left ( T_{oil,out} - T_{water,out} \over T_{oil,in} - T_{water,in} \right )} \\ = {(\SI{450}{\kelvin} - \SI{400}{\kelvin}) - (\SI{550}{\kelvin} - \SI{300}{\kelvin}) \over ln \left (\SI{450}{\kelvin} - \SI{400}{\kelvin} \over \SI{550}{\kelvin} - \SI{300}{\kelvin} \right )} \\ = \SI{124.3}{\kelvin}$$