Answer:
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
