a closed box has a fixed surface area A and a suqare base wide side X

Question

a) find the formula for its volume, v, as a function x
b) sketch the graph of v against x
c) find the max value of V

Answer 1

Let the height be y
then A = 2x^2 + 4xy
4xy = A - 2x^2
y = (A - 2x^2)/(4x), remember A is a constant.

a) Volume = x^2y
= x^2(A - 2x^2)/(4x)
= (Ax - 2x^3)/4

b) we can't do sketching on here

c) V = (Ax - 2x^3)/4
dV/dx = A/4 - (3/2)x^2 = 0 for max of V
(3/2)x^2 = A/4
x^2 = A/6
x = √(A/6) or √A/√6

V = [(A(√A/√6) - 2(A/6)(√A/√6)]/4
= A√A/(6√6)

Check my algebra, I should have written it down on paper.
If in your answer key you don't see a radical in the bottom, they might have rationalized the denominator.