Question

The height of a cone is decreasing at a constant rate of 3 centimeters per minute. The volume remains a constant 52 cubic centimeters. At the instant when the radius of the cone is 66 centimeters, what is the rate of change of the radius?

Answer 1

The rate of change of the radius of the cone is 3 centimeters per minute.

Step-by-step explanation:

To find the rate of change of the radius, we need to determine the relationship between the radius and height of the cone. The volume of a cone is given by the formula V = (1/3)πr^2h, where V is the volume, r is the radius, and h is the height. We are told that the volume is constant at 52 cubic centimeters, so we have:

(1/3)πr^2h = 52

Since the height is decreasing at a constant rate of 3 centimeters per minute, we can express h as a function of time t as h = h0 - 3t, where h0 is the initial height. Substituting this into the equation, we get:

(1/3)πr^2(h0 - 3t) = 52

Simplifying, we have πr^2(h0 - 3t) = 156

Now, we can differentiate both sides of the equation with respect to time t:

d(πr^2(h0 - 3t))/dt = d(156)/dt

Using the product and chain rule, the left side becomes:

2πr(dh0/dt - 3) = 0

We are given that when the radius is 66 centimeters, so r = 66, we want to find the rate of change of the radius, so we want to find dr/dt. Plugging in the values into the equation, we get:

2π(66)(dh0/dt - 3) = 0

Dividing both sides by 2π(66), we have:

dh0/dt - 3 = 0

Adding 3 to both sides, we get:

dh0/dt = 3

Therefore, the rate of change of the radius is 3 centimeters per minute.

Answer 2

The rate of change of the radius is 0 centimeters per minute.

Step-by-step explanation:

To find the rate of change of the radius, let's use the formula for the volume of a cone: V = (1/3) * π * r^2 * h, where V is the volume, r is the radius, and h is the height. Since the volume remains constant at 52 cubic centimeters, we can substitute the values into the formula and solve for the height: (1/3) * π * r^2 * h = 52. Substituting the given radius of 66 centimeters, we get (1/3) * π * (66)^2 * h = 52. Solving for h, we find that the height is approximately 0.635 centimeters. To find the rate of change of the radius, we can use the related rates formula: dV/dt = (dV/dr) * (dr/dt), where dV/dt is the rate of change of the volume, dV/dr is the partial derivative of the volume with respect to the radius, and dr/dt is the rate of change of the radius.

Since the volume is constant and the height is decreasing at a rate of 3 centimeters per minute, the rate of change of the volume, dV/dt, is 0. Therefore, 0 = (dV/dr) * (dr/dt). Solving for dr/dt, we find that the rate of change of the radius is 0 centimeters per minute. This means that the radius remains constant at 66 centimeters, regardless of the changing height.