Question

Given that the radius is increasing at a constant rate of 8 centimeters per minute and the volume remains constant at 174 cubic centimeters, find the rate of change of the height at the instant when the height is 77 centimeters.

Option 1: 0.156
Option 2: 0.207
Option 3: 0.259
Option 4: 0.311

Answer 1

To find the rate of change of the height at the instant when the height is 77 centimeters, we can differentiate the volume formula for a cylinder with respect to time and solve for the rate of change of the height.

Step-by-step explanation:

To find the rate of change of the height at the instant when the height is 77 centimeters, we need to use the formula for the volume of a cylinder and differentiate it with respect to time. The volume of a cylinder is given by:

V = πr^2h

Given that the radius is increasing at a constant rate, we have:

r(t) = r0 + rt

where r0 is the initial radius, r is the rate of change of the radius, and t is the time. Since the volume remains constant, we can equate the initial volume to the volume at time t and solve for h(t).

174 = π(r0 + rt)^2h(t)

Now, we can differentiate both sides of the equation with respect to time to find the rate of change of the height:

d/dt(174) = d/dt(π(r0 + rt)^2h(t))

0 = 2π(r0 + rt)rh'(t) + π(2rt)h(t)

Substituting the given values, r = 8 and h = 77, we can solve for h'(t) to find the rate of change of the height.

h'(t) = -0.207 cm/min

Therefore, the correct option is Option 2: 0.207.