the radius of the right circular cylinder shown below is growing at a rate of 2ft/min while it's height is shrinking at 3ft/min. At what rate is the volume of the cylinder changing, with respect to time, when the radius is 4ft and the volume is 32 ft cubed.

The volume is decreasing at a rate of about 118.8 cubic feet per minute.

Step-by-step explanation:

Recall that the volume of a cylinder is given by:

Take the derivative of the equation with respect to t. V, r, and h are all functions of t:

Use the product rule and implicitly differentiate. Hence:

We want to find the rate at which the volume of the cylinder is changing when the radius if 4 feet and the volume is 32 cubic feet given that the radius is growing at a rate of 2ft/min and the height is shrinking at a rate of 3ft/min.

In other words, we want to find dV/dt when r = 4, V = 32, dr/dt = 2, and dh/dt = -3.

Since V = 32 and r = 4, solve for the height:

Substitute:

Therefore, the volume is decreasing at a rate of about 118.8 cubic feet per minute.

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