An open–top rectangular tank is to be constructed so that its length is twice its width

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

An open–top rectangular tank is to be constructed so that its length is twice its width. Let

w

(in meters) denote the width of the base and

h

(in meters) denote the depth of the tank. The tank must hold exactly

10\text{ m}^3

of water. (a) Express the interior surface area

A(w,h)

(including the base and four sides but excluding the top) in terms of

w

and

h

. (b) Form the ratio

R(w,h)=\dfrac{A(w,h)}{10}

and, using the volume constraint, rewrite

R

entirely as a function of

w

. (c) Compute

\dfrac{dR}{dw}

, set the derivative equal to zero, and solve for all positive critical values of

w

. Provide exact algebraic expressions before giving decimal approximations to three significant figures. (d) Show that the critical value discovered in part (c) gives an absolute minimum of

R

. Then state the optimal dimensions

(w,2w,h)

of the tank and the minimized surface–to–volume ratio, each to three significant figures.

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