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

Length and Width = 10ft

Height = 5ft

Surface Area = 300 square feet

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 = (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*(500)/(LW)(L + W)`

`S = L * W + (1000)/(LW)(L + W)`

`S = L W + (1000)/(L) + (1000)/(W)`

Differentiate with respect to L and to W

`S'(W) = L - (1000)/(W^2)`

and

`S'(L) = W - (1000)/(L^2)`

Equate both to get the critical value

`S'(W) = L - (1000)/(W^2)`and
`S'(L) = W - (1000)/(L^2)`

`0 = L - (1000)/(W^2)` and
`0 = W - (1000)/(L^2)`

`(1000)/(W^2) = L` and
`(1000)/(L^2) = W`

`W^2L = 1000` and
`L^2W = 1000`

Make L the subject in
`W^2L = 1000`

`L = (1000)/(W^2)`

Substitute
`(1000)/(W^2)` for L in
`L^2W = 1000`

`((1000)/(W^2))^2 * W = 1000`

`(1000000)/(W^4) * W = 1000`

`(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 = (1000)/(W^2)`

`L = (1000)/(10^2)`

`L = (1000)/(100)`

`L = 10`

Recall that:
`H = (500)/(LW)`

`H = (500)/(10*10)`

`H = (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`