FE Civil Mathematics and Statistics Practice Problems

The current FE Civil specification assigns 8-12 of the 110 exam questions to Mathematics and Statistics. Form A contains 9 original problems in this area. The sample below is published in full, with the same worked-solution format used throughout the book.

A rectangular open channel must provide a flow cross-sectional area of $8 m^{2}$ . The lining covers the channel bottom and both vertical sides; the water surface is free. The minimum lined perimeter of the cross section is most nearly:

(A)6.0 m
(B)8.0 m
(C)8.5 m
(D)12.0 m

Show worked solution

Answer: (B)

Lining is minimized at the half-square section: $d = 2$ m, $b = 4$ m, giving $P = b + 2 d = 8.0$ m.

With bottom width $b$ and depth $d$ , the constraint $b d = 8$ makes the lined perimeter a function of depth alone.

P = b + 2 d = \frac{8}{d} + 2 d

The optimum lies where the slope of $P (d)$ vanishes.

\frac{d P}{dd} = - \frac{8}{d ^{2}} + 2 = 0

d^{2} = 4, d = 2 m

Then $b = 8/2 = 4$ m, and $16/ d^{3} > 0$ confirms a minimum.

P = 4 + 2 (2) = 8.0 m

FE Reference Handbook — Mathematics: Differential Calculus, Test for a Minimum

Why the other choices appear

(A)Sums one width and one side wall only, $b + d = 4 + 2 = 6.0$ m, omitting the second side.
(C)Assumes a square section with $b = d = 8 = 2.83$ m, giving $P = 3 (2.83) = 8.5$ m.
(D)Counts the free water surface as lined, $2 b + 2 d = 8 + 4 = 12.0$ m.