Question

A constant volume of pizza dough is formed into a cylinder with a relatively small height and large radius. The dough is spun and tossed into the air in such a way that the height of the dough increases, but its cylindrical shape is maintained. At a certain time, the height of the dough is 12 inches, and the radius of the dough is increasing at a rate of 2 inches per second. Find the numerical answer and units for the rate of change of the volume of the dough with respect to time.

A. 48π cubic inches per second
B. 24π cubic inches per second
C. 96π cubic inches per second
D. 64π cubic inches per second

Answer 1

Without the initial radius of the pizza dough, we cannot calculate a numerical value for the rate of change of its volume as the radius changes. The rate of change of volume is related to the radius and the given height, but more information is needed to find a specific numerical answer.

Step-by-step explanation:

The student wants to find the rate of change of the volume of the pizza dough as it's spun into the air with a changing radius. When the height of the dough is 12 inches, the radius is increasing at a rate of 2 inches per second. The volume of a cylinder is given by V = πr²h. Given that the volume is a function of the radius, we can use the chain rule to differentiate with respect to time t:

Since dV/dt = dV/dr × dr/dt, and given that dr/dt = 2 in/s, we can substitute to find the rate of change of the volume. The area of the cylinder's base A is πr². Differentiating, we get 2πrh. Multiplying this by dr/dt, we find:

dV/dt = 2πr(12) × 2, which simplifies to dV/dt = 48πr. Since we are given no initial radius, we cannot calculate a numerical value without it. Therefore, based on the information provided, a numerical answer cannot be determined.