A rectangular solid with a square base has a surface area of 37.5 square centimeters. (Let w represent the width of the sides of the square base and let h represent the height of the solid.)

Question

A rectangular solid with a square base has a surface area of 37.5 square centimeters. (Let w represent the width of the sides of the square base and let h represent the height of the solid.)

(a)Determine the dimensions (in cm) that yield the maximum volume.

w=

h=

(b)Find the maximum volume (in cm³).

 

 

Answer 1

Surface area = 4hw + 2w² = 37.5

h = (37.5 - 2w²)/(4w)

Volume = V = hw³ = (37.5w - 2w²)/4

dV/dw = 37.5/4 -3w²/2

Setting dV/dw to zero and solving for w:

w = √((37.5/4)(2/3)) = 2.5 cm

h = (37.5 - 2(2.5)²)/10 = 2.5 cm (a perfect cube)

Max volume = (2.5)³ = 15.625 cm³

Checking for surface area, A = 6(2.5)² = 37.5