Calculus Problem: Please Help!!

Question

A rectangular box has a square base with an edge length of x cm and a height of h cm. The volume of the box is given by V = x²h cm³. Find the rate at which the volume of the box is changing when the edge length of the base is 4 cm, the edge length of the base is increasing at a rate of 2 cm/min, the height of the box is 15 cm, and the height is decreasing at a rate of 3 cm/min.

Options:

1. The volume of the box is decreasing at a rate of 192 cm³/min.
2. The volume of the box is increasing at a rate of 288 cm³/min.
3. The volume of the box is decreasing at a rate of 288 cm³/min.
4. The volume of the box is increasing at a rate of 192 cm³/min.

Answer 1

Since V = x²h and BOTH x and h are changing with time:

dV/dt = 2xh(dx/dt) + x²(dh/dt).

Now just plug in the numbers to answer the question.