FE Math Practice Problems — Free Worked Answer

Work 3 free FE Mechanical mathematics practice problems with complete solutions, governing equations and checks. Attempt the setup before reading the answer.

Free sample problem

A right circular conical hopper stands apex down, $8$ ft deep with an $8$ ft top diameter. Granular material is fed in at a steady $12$ ft $^3$ /min. When the material depth is $6$ ft, the rate at which the surface level rises is most nearly:

(A)0.106 ft/min
(B)0.141 ft/min
(C)0.424 ft/min
(D)1.273 ft/min

Show worked solution

Answer: (C)

Because the free surface at 6 ft depth spans only 28.3 ft $^2$ , a 12 ft $^3$ /min feed lifts the level about 0.42 ft/min.

Similar triangles fix the surface radius at half the depth, which collapses the cone volume to a function of depth alone.

r = \frac{h}{2}

V = \frac{\pi}{3}r^2 h = \frac{\pi h^3}{12}

Differentiating in time links the two rates, and the instantaneous depth of 6 ft closes the problem.

\frac{dV}{dt} = \frac{\pi h^2}{4}\cdot\frac{dh}{dt}

\frac{dh}{dt} = \frac{4(12)}{\pi (6)^2} = 0.424 \text{ ft/min}

FE Reference Handbook — Mathematics: Derivatives and Indefinite Integrals; Mensuration of Areas and Volumes, Right Circular Cone

Why the other choices appear

(A)Taking $r = h$ instead of $h/2$ gives $dh/dt = 12/[\pi(6)^2] = 0.106$ ft/min.
(B)Dropping the $1/3$ from the cone volume gives $V = \pi h^3/4$ and $dh/dt = 48/[3\pi(6)^2] = 0.141$ ft/min.
(D)Carrying the $1/3$ into the rate equation as $dV/dt = (\pi/3)r^2\,dh/dt$ with $r = 3$ ft gives $12/9.425 = 1.273$ ft/min.

Two more free problem types

Try a different setup before reading the solution. These additional original questions are separate from the sold practice forms, so solving them does not reveal another question from a full practice form.

Separate an exponential model from its time constant

A temperature model is $T(t)=25+50e^{-kt}$ , with T in degrees C and t in minutes. A measurement gives T(10) = 45. Under this model, at what time after t = 0 does T reach 35 degrees C?

(A)6.44 min
(B)10.00 min
(C)17.56 min
(D)25.00 min

Show worked solution

Answer: (C)

The model reaches 35 degrees C at about 17.56 minutes.

The excess temperature above the 25-degree surroundings falls from 50 to 20 in ten minutes. Solve for the rate parameter before extrapolating.

e^{-10k}=20/50,\qquad k=\ln(2.5)/10=0.091629\ \mathrm{min}^{-1}

At 35 degrees C the excess temperature is 10, so the exponential ratio is one fifth.

t=\frac{\ln(50/10)}{k}=17.5647\ \mathrm{min}

Substitute the time into the given model: 25 + 50 exp(-k t) = 35. The time exceeds ten minutes, as it must for a lower temperature.

FE Reference Handbook: Mathematics — exponential functions and first-order differential equations.

Why the other choices appear

(A)This reverses the logarithm ratio.
(B)The stated ten-minute observation is 45 degrees, not 35.
(D)This assumes a linear cooling rate instead of exponential decay.

Numerical integration from measured flow

A pump flow rate is measured at t = 0, 2, and 4 minutes as 2, 6, and 4 cubic metres per minute, respectively. Using the composite trapezoidal rule with these two equal intervals, what volume is pumped from t = 0 to t = 4 minutes?

(A)12 m³
(B)16 m³
(C)18 m³
(D)24 m³

Show worked solution

Answer: (C)

The trapezoidal estimate is 18 cubic metres.

Integrate flow rate over time. Each two-minute interval uses the average of its endpoint rates.

V\approx\frac{2}{2}(2+6)+\frac{2}{2}(6+4)=8+10=18\ \mathrm{m}^3

In the composite form, interior readings receive twice the endpoint weight.

V\approx\frac{h}{2}(q_0+2q_1+q_2)=\frac{2}{2}(2+12+4)

The average rate over all four minutes is 4.5 cubic metres per minute. Multiplying by four independently recovers 18 cubic metres. This is a numerical estimate; it is exact only if the rate varies linearly within each interval.

FE Reference Handbook: Mathematics — numerical integration, trapezoidal rule.

Why the other choices appear

(A)This adds the three rates without incorporating time intervals.
(B)This uses only the two endpoints and discards the middle reading.
(D)This applies the maximum rate for all four minutes.

Download all 14 additional questions, diagrams and complete solutions (free PDF). Download the editable practice review sheet (CSV).

Original questions and diagrams by EngineerProof, September 30, 2026. Personal study and unchanged redistribution with attribution are permitted. The questions are independent practice, not official NCEES material. Report a correction.

Three free problem types · 14 more across two forms

What FE math practice problems cover

The current NCEES FE Mechanical specification, effective July 2020, assigns 6–9 of the 110 questions to Mathematics. The listed scope is analytic geometry, calculus, ordinary differential equations, linear algebra, numerical methods, and algorithm and logic development. The free related-rates question above tests calculus and geometry together; it is not a review of every listed topic.

NCEES provides the current FE Reference Handbook through its FE exam resource page. Use that searchable electronic reference while practicing, but do not treat a formula search as the whole solution: the hopper problem requires the similar-triangle relation before the volume equation can be differentiated.

A three-step check for the free problem

Reduce the geometry first. The 8-foot depth and 4-foot top radius give r = h/2, so the cone volume becomes a function of depth alone.
Differentiate rates, not snapshots. Differentiate V = πh³/12 with respect to time, then use the stated 12 ft³/min inflow and 6-foot depth.
Check the physical result. The positive 0.424 ft/min answer also equals flow divided by the 28.27-ft² free-surface area; that independent check rejects the three displayed distractors.

Free math problem and full practice forms

Resource	Mathematics practice	Complete release	Price
This HTML page	3 complete questions: related rates, exponential modeling, numerical integration	Worked solution, independent check, and all three distractors	USD 0
Free FE Mechanical sample PDF	The same 1 Mathematics problem	18 pages · 14 worked problems, one per Mechanical area	USD 0
FE Mechanical Form A	7 Mathematics questions	110 questions · 129-page PDF · complete solutions and area score sheet	USD 29
FE Mechanical Form B	7 Mathematics questions	110 questions · 107-page PDF · complete solutions and area score sheet	USD 29

Forms A and B are sold separately and contain different questions. They are independent printable practice PDFs, not official NCEES questions or a reproduction of the computer-based interface. See actual PDF previews and compare both forms. Practice questions and solutions are by EngineerProof. Editorial method and corrections.

Free FE Mechanical problems in the other knowledge areas

Probability and Statistics · Ethics and Professional Practice · Engineering Economics · Electricity and Magnetism · Statics · Dynamics, Kinematics, and Vibrations · Mechanics of Materials · Material Properties and Processing · Fluid Mechanics · Thermodynamics · Heat Transfer · Measurements, Instrumentation, and Controls · Mechanical Design and Analysis

Retaking the FE exam? The FE Mechanical retake plan turns the NCEES diagnostic report into an area-by-area plan, starting from the 37 percent repeat pass rate. The same guide covers testing windows, the attempt limit, and the registration fee.