Question

The radius of a cylinder is increasing at a constant rate of 8 centimeters per minute. The volume remains a constant 174 cubic centimeters. At the instant when the height of the cylinder is 7 centimeters, what is the rate of change of the height? The volume of a cylinder can be found with the equation V=πr²h. Round your answer to three decimal places. Answer:___cm/min

Answer 1

The rate of change of the height of the cylinder at the instant when the height is 7 centimeters is approximately -4.777 cm/min.

Step-by-step explanation:

To find the rate of change of the height of the cylinder, we can use the formula for the volume of a cylinder: V = πr²h.

Since the volume remains constant at 174 cubic centimeters, we can write the equation as 174 = π(r²)(h).

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

dV/dt = 0 = 0 + π(2r)(dr/dt)(h) + π(r²)(dh/dt).

Given that the radius is increasing at a constant rate of 8 centimeters per minute, dr/dt = 8. We can substitute this into the equation and solve for dh/dt:

0 = π(2r)(8)(7) + π(r²)(dh/dt).

simplifying, we get dh/dt = -112/(πr²).

Substituting the radius r = 0.75 cm, we find dh/dt ≈ -4.777 cm/min.