(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.

Length and Width = 10ft

Height = 5ft

Surface Area = 300 square feet

Step-by-step explanation:

Given

-- Volume

Let:

Volume (V) is calculated as:

Substitute 500 for V

Make H the subject

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

Substitute 500/LW for H

Differentiate with respect to L and to W

and

Equate both to get the critical value

and

and

and

and

Make L the subject in  

Substitute for L in

Cross Multiply

Divide both sides by 1000

Take cube roots of both sides

Substitute 10 for W in

Recall that:

So, the dimensions are:

and

The surface area is:

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step-by-step explanation: