Final answer:
The rate of change of volume when the piston is 14 cm from the base of the cylinder is 1078π cm3/s. This is calculated by taking the derivative of the cylinder's volume with respect to time and applying the given values for radius and height.
Step-by-step explanation:
The student is asking about the rate of change of the volume of a cylinder as a piston moves into it. To solve this problem, we need to find the derivative of the volume with respect to time (dv/dt). The Volume of the cylinder (V) is given by the formula V = πr2h, where r is the radius and h is the height. As the piston moves, it changes the height of the cylinder at a constant speed. Here, r = 7 cm, and the piston speed, dh/dt, is 7 cm/s. When the piston is 14 cm from the base, h remains constant; therefore, we are only interested in the derivative at this instant.
Using the formula for the volume of the cylinder, the derivative with respect to time is dV/dt = π(72)(7) = 1078π cm3/s. Therefore, the correct answer is (c) 1078π cm3/s.