Question

The radius of a cylinder is decreasing at a constant rate of 9 meters per minute. The volume remains a constant 728 cubic meters. At the instant when the height of the cylinder is 3 3 meters, what is

Answer 1

The question is about finding the relationship between the changing radius and height of a cylinder with constant volume, using calculus and the formula for the volume of a cylinder, V = πr²h.

Step-by-step explanation:

The student's question deals with the rate of change in the radius of a cylinder and its relation to the volume and height of the cylinder. Given that the volume of the cylinder remains constant at 728 cubic meters, and the height is currently 3.3 meters, the goal is to understand how these dimensions affect each other dynamically over time, using calculus or related mathematical principles to determine various rates of change.

To answer this question, one would typically use the formula for the volume of a cylinder, V = πr²h, where V is volume, r is radius, and h is height. This formula can be rearranged and differentiated with respect to time to find the rate at which the height changes as the radius decreases. This is a real-world application of related rates, a topic covered in calculus. However, without the specific rate-of-change question stated, a complete solution cannot be provided.

Answer 2

The question involves calculus and related rates of change to determine the rate at which the height of a cylinder changes. The constant volume of the cylinder allows us to set the derivative of the volume formula with respect to time to zero and solve for the height change rate.

Step-by-step explanation:

The question is asking about the rate at which the height of a cylinder is changing given that the radius is decreasing and the volume of the cylinder remains constant at 728 cubic meters. This is a calculus problem involving related rates of change. To find the rate of change of the height, we can use the formula for the volume of a cylinder V = πr²h, where V represents the volume, r is the radius, and h is the height. Since the volume is constant, we can differentiate this formula with respect to time to find the relationship between the rates of change of the radius and height.

At the instant when the height is 3.3 meters, we need to set the derivative of the volume with respect to time to zero and solve for the rate of change of the height as the radius decreases. Without the actual rates given, the problem cannot be solved numerically. However, this would typically involve taking the derivative of the formula dV/dt = π(2r • dr/dt)h + πr² • dh/dt, setting dV/dt to zero, and solving for dh/dt using the given rate of change of the radius dr/dt.