A rectangular box open at the top has a square base. The internal side of the base is x cm long and the total internal surface area of the box is 432cm2.

Question

Express in terms of x.
(i) the internal height h, of the box;

(ii) the internal volume V, of the box.

(b) Find:

i) the value of x for which the volume V is maximum;

(ii) the maximum internal volume of the box.

Answer 1

4x^2h = 432, so h = 108/x^2
v = x^2h = 108h

dv/dx = 2xh + x^2 dh/dx = 2xh - 216/x
now find where dv/dx = 0 for max v.

Answer 2

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