Final answer:
The rate of change of the volume of a cylinder when its radius is changing is found by differentiating the volume formula with respect to time and substituting the rate of radius change, given constant height.
Step-by-step explanation:
The student has asked to find the rate of change of the volume of a cylinder when its radius is changing. To do this, we need to use calculus, specifically the concept of related rates. The formula for the volume of a cylinder is V = πr²h, where V is the volume, r is the radius, and h is the height. To find the rate of change of the volume with respect to the radius, we differentiate both sides of this equation with respect to time t.
Steps:
- Write the volume formula: V = πr²h.
- Differentiate both sides with respect to t: dV/dt = π(2rh dr/dt + r² dh/dt).
- If the height h is constant, the equation simplifies to dV/dt = 2πrh dr/dt.
- To find the rate of change of volume volume change, substitute the given rate of change of the radius dr/dt along with the values of r and h into the equation.
If the height is not constant, you would need to know the rate of change of height as well, dh/dt, to find the total rate of change of volume.