Question

Recall that the volume V of a cylinder with radius r and height h is V = πr²h. Sally places a marshmallow in the microwave to make a S'more. Assume that the marshmallow has the shape of a cylinder with a radius of 1 cm and a height of 3 cm. When she turns on the microwave, the marshmallow starts to expand at the rate of 2 cm per second. Assume also that as it grows, the height is always three times the radius. Find the rate at which the radius is increasing when the radius is 4 cm per second.

A. 2 cm per second
B. 1 cm per second
C. 1.5 cm per second
D. 3 cm per second
E. 4 cm per second

Answer 1

The rate at which the radius of the cylindrical marshmallow is increasing when the radius is 4 cm, given the relationship between height and radius (height being three times the radius), is 1 cm per second.

Step-by-step explanation:

The student asked about the rate at which the radius of a cylindrical marshmallow is increasing when the radius is 4 cm, given that its height is always three times its radius. Using the volume formula V = πr²h, we can derive the relation between the radius r and the height h of the cylinder, which is h = 3r. Since the marshmallow's volume increases uniformly as it expands, we can use differentiation to find the rate of change. Differentiating both sides of the volume equation with respect to time t, we get dV/dt = πr² (dh/dt) + 2πrh (dr/dt). Substituting h = 3r and dh/dt = 3(dr/dt) and given that the volume increases at a rate of 2 cm³ per second (dV/dt) when the radius is 4 cm, we can now solve for dr/dt. This gives us the rate at which the radius is increasing when the radius is 4 cm, which can be calculated to be 1 cm per second, making option B the correct answer.