Question

(Bonus) A rectangular tank with a bottom and sides but no top is to have volume 500 cubic feet. Determine the dimensions (length, width, height) with the smallest possible surface area.

Answer 1

Answer:

Length and Width = 10ft

Height = 5ft

Surface Area = 300 square feet

Step-by-step explanation:

Given

$V = 500ft^3$ -- Volume

Let:

$L = Length$

$W =Width$

$H = Height$

Volume (V) is calculated as:

$V = L * W * H$

Substitute 500 for V

$500 = L * W * H$

Make H the subject

$H = \frac{500}{LW}$

The tank has no top. So, the surface area (S) is:  

$S = L * W + 2*H*L + 2*H*W$

$S = L * W + 2H(L + W)$

Substitute 500/LW for H

$S = L * W + 2*\frac{500}{LW}(L + W)$

$S = L * W + \frac{1000}{LW}(L + W)$

$S = L W + \frac{1000}{L} + \frac{1000}{W}$

Differentiate with respect to L and to W

$S'(W) = L - \frac{1000}{W^2}$

and

$S'(L) = W - \frac{1000}{L^2}$

Equate both to get the critical value

$S'(W) = L - \frac{1000}{W^2}$and $S'(L) = W - \frac{1000}{L^2}$

$0 = L - \frac{1000}{W^2}$ and $0 = W - \frac{1000}{L^2}$

$\frac{1000}{W^2} = L$ and $\frac{1000}{L^2} = W$

$W^2L = 1000$ and $L^2W = 1000$

Make L the subject in  $W^2L = 1000$

$L = \frac{1000}{W^2}$

Substitute $\frac{1000}{W^2}$ for L in $L^2W = 1000$

$(\frac{1000}{W^2})^2 * W = 1000$

$\frac{1000000}{W^4} * W = 1000$

$\frac{1000000}{W^3} = 1000$

Cross Multiply

$1000000 = 1000W^3$

Divide both sides by 1000

$1000 = W^3$

Take cube roots of both sides

$\sqrt[3]{1000} = W$

$10 = W$

$W = 10$

Substitute 10 for W in $L = \frac{1000}{W^2}$

$L = \frac{1000}{10^2}$

$L = \frac{1000}{100}$

$L = 10$

Recall that:$H = \frac{500}{LW}$

$H = \frac{500}{10*10}$

$H = \frac{500}{100}$

$H = 5$

So, the dimensions are:

$L, W=10$ and $H = 5$

The surface area is:

$S = L * W + 2H(L + W)$

$S = 10*10 +2*5(10+10)$

$S = 10*10 +2*5*20$

$S = 100 + 200$

$S = 300$