If 2300 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box. Volume

Step-by-step explanation:

Let the volume of the box be V = x²h

x is the length of one side of the square base

h is the height of the box;

Also let the surface area S = x² + 2xh + 2xh (open at the top)

S = x²+4xh

Given

S = 2300sq.cm

2300 = x²+4xh

make h the subject of the formula;

2300-x² = 4xh

h = (2300-x²)/4x

Substitute this expression into the volume formula

V = x²y

V = x²(2300-x²)/4x

V = x(2300-x²)/4

V = 1/4(2300x - x³)

To maximize the volume, dV/dx = 0

dV/dx = 1/4(2300- 3x²)

1/4(2300- 3x²) = 0

cross multiply

2300-3x² = 0

2300 = 3x²

x² = 2300/3

x² = 766.67

x = √766.67

x = 22.69cm

Since h = (2300-x²)/4x

h = 2300 - 766.67/4(22.69)

h = 1533.33/90.76

h = 16.89cm

Volume = 766.67(16.89)

Volume = 12,949.0563cm³

Hence the largest possible volume of the box is 12,949.0563cm³