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

Question

(a) Find the dimensions 𝑥, ℎ for which the volume is 24 and the surface area is as small as possible. (Use symbolic notation and fractions where needed.)

𝑥=

ℎ=

(b) Find the dimensions 𝑥, ℎ for which the surface area is 7 and the volume is as large as possible. (Use symbolic notation and fractions where needed.)

x=

h=

Answer 1

V = hx^2 = 24

boxes which are rectangular prisms

have least surface area for a fixed volume when they are cubes

h=x=cube root of 24 = about 2.8845

or use calculus to find the optimal dimensions

h=24/x^2

A = area of bottom + area of the top + 4 times the area of a side

= 2x^2 + 4xh

= 2x^2 + 4x(24/x^2)

= 2x^2 + 96/x

take the derivative and set equal to zero

A'(x) = 4x -96/x^3= 0

4x^2 -96 = 0

x^3 = 96/4 = 24

x = cube root of 24 = about 2.8845

h=24/x^2 = 24/(2.8845^2) = about 2.8845

a cube has the least surface area for a given volume

surface area = 7

find the largest volume

it's a cube again that maximizes volume for a fixed surface area

surface area = 4xh +2x^2, with x=h

or area = 4x^2 +2x^2 = 6x^2 = 7

x^2=7/6

x =sqr(7/6) = about 1.08 = x = h

largest volume = 1.08^3 = about 1.26

surface area = 2(1.08)^2 + 4(1.08)^2 = 6(1.08^2) =7

or use calculus to find the optimal dimensions

and get the same answers x=about 1.08 = h