Piston compression A piston is seated at the top of a cylindrical chamber with radius 5 cm when it starts moving into the chamber at a constant speed of 3 cm/s (see figure). What is the rate of change of the volume of the cylinder when the piston is 2 cm from the base of the chamber? <IMAGE>

Verified step by step guidance

1. Identify the formula for the volume of a cylinder, which is given by V = πr²h, where r is the radius and h is the height of the cylinder.
2. Substitute the known radius of the cylinder (5 cm) into the volume formula to express the volume in terms of the height (h): V = π(5)²h = 25πh.
3. Determine the relationship between the height of the cylinder and the position of the piston. Since the piston is moving into the chamber, the height h can be expressed as h = H - d, where H is the total height of the chamber and d is the distance of the piston from the base.
4. Differentiate the volume V with respect to time t using the chain rule: dV/dt = dV/dh * dh/dt. Here, dV/dh is the derivative of the volume with respect to height, and dh/dt is the rate at which the height is changing, which is negative since the piston is moving down.
5. Evaluate dV/dh at the height when the piston is 2 cm from the base and multiply it by the rate of change of height (dh/dt) to find the rate of change of the volume dV/dt.