Dimensions and Maximal Volume

Question

Find the dimension of a box with a square base with surface area 20 and the maximal volume.

What is the side of the base, height, and maximum volume?

Answer 1

s represents the sides of the square base and top

h represents the height of the box

20 = 2s² + 4sh

10 = s² + 2sh

10 - s² = 2sh

(10 - s²) / 2s = h

V = s²h

V = s²[(10 - s²) / 2s]

V = (10s² - s⁴) / 2s

V = (10s - s³) / 2

V = 5s - (0.5)s³

V' = 5 - 1.5s²

0 = 5 - 1.5s²

1.5s² = 5

s² = 5 / 1.5

s = √7.5 / 1.5

(10 - s²) / 2s = h

(10 - (5/1.5) / [2(√7.5/1.5) = h

√7.5 / 1.5 = h

QED

Answer 2

a cube has maximum volume

surface area of a cube is 6 times the side length squared: 6s^2 = 20. s^2 = 20/6= 10/3. . s= sqr(10/3)= about 1.826

maximum volume = s^3 = sqr(10/3)^3 = (10/3)sqr(10/3) = about 6.1 cubic units

dimensions are about 1.826 x 1.826 x 1.826