A box has a square base of side x and height y.

Question

(a) Find the dimensions x, y for which the volume is 2 and the surface area is as small as possible. 

(b) Find the dimensions for which the surface area is 7 and the volume is as large as possible. 

Answer 1

Volume = 2 = x²y

and y=2/x²

Surface area = 2x² +4xy = 2x² +2/x²

take the derivative and set it equal to zero, then solve for x

to minimize surface area

4x -4/x³ = 0 multiply by x cubed

4x⁴ -4 = 0

x⁴ = 1

x= 1

y=2