Final answer:
The rate of change of the volume of a cylinder with a fixed radius of 4 cm and a height increasing at 2 cm/min is 16π cm³/min when the height is 14 cm.
Step-by-step explanation:
The question is related to the rate of change of the volume of a cylinder. The volume V of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height of the cylinder. Here, we have a cylinder with a fixed radius of 4 cm whose height is increasing at the rate of 2 cm/min. We want to find the rate at which the volume of the cylinder is increasing when the height is 14 cm.
To solve this, we use the concept of related rates in calculus. Differentiating the volume with respect to time t, we get dV/dt = π(2r)(dh/dt), because the radius is constant. Plugging in the values r = 4 cm and dh/dt = 2 cm/min, we can calculate the rate of change of volume.
dV/dt = π(2 × 4 cm)(2 cm/min)
dV/dt = 16π cm^2/min
Therefore, the rate of change of the volume of the cylinder with respect to time is 16π cubic centimeters per minute when the height is 14 cm.