Answer:
500
Step-by-step explanation:
It's a box with a square base, so let's say the width and length are x and the height is y.
The surface area of the box without the top is:
A = x² + 4xy
300 = x² + 4xy
The volume of the box is:
V = x²y
Solve for y in the first equation and substitute into the second:
y = (300 − x²) / 4x
V = x² (300 − x²) / 4x
V = x (300 − x²) / 4
V = 75x − ¼ x³
To optimize V, find dV/dx and set to 0:
dV/dx = 75 − ¾ x²
0 = 75 − ¾ x²
x = 10
So the volume of the box is:
V = 75x − ¼ x³
V = 500
The maximum volume is 500 cm³.
