Optimization problem

Question

A box with a square base and an open top will be made from 1400 cm^2 of material.

Find the largest possible volume of the box.

Volume=???

Answer 1

SA = Surface Area of the open box

V = volume of the open box

x = length of the sides of the base of the box

SA = 1400 cm²

SA = x² + 4xh

1400 = x² + 4xh

h = (1400 - x²)/4x

V = x²h

V = x²[(1400 - x²)/4x]

V = (x/4)(1400 - x²)

dV/dx = (x/4)(-2x) + (1/4)(1400 - x²)

Set dV/dx = 0

(x/4)(-2x) + (1/4)(1400 - x²) = 0 Product rule

(-1/2)x² + 350 - (1/4)x² = 0

(-3/4)x² + 350 = 0

(3/4)x² = 350

(4/3)(3/4)x² = 350(4/3)

x² = 1400/3

x = √1400/√3

x = (10√42)/3 ≈ 21.602 cm

h = (1400 - x²)/4x

V = (x/4)(1400 - x²)

V = (1/4)(√1400/√3)(1400 - (1400/3))

= (1/4)(√100•√14/√3)(2800/3)

= (1/4)(10√42/3)(2800/3)

= 7000√42/9

= 5,040.57609876167

≈ 5,040.576 cm³

The largest possible volume is 5,040.576 cm³