Final answer:
To find the rate of increase in the volume of a sphere, we need the rate at which the radius is changing.
Step-by-step explanation:
First, let's find the formula for the volume of a sphere. The formula is V = (4/3)πr³, where V is the volume and r is the radius of the sphere.
To find the rate of increase in volume, we need to find the derivative of the volume formula with respect to time. Let's denote the rate of increase in volume as dv/dt.
Now, dv/dt = dV/dt = d((4/3)πr³)/dt
We can differentiate the formula using the power rule for derivatives: d(xⁿ)/dt = n*xⁿ⁻¹*dx/dt. Applying this rule, we have:
dv/dt = d((4/3)πr³)/dt = (4/3)π*3r² * dr/dt
Substituting the given values, when the radius is 2 cm, r = 2 cm, and when the radius is 5 cm, r = 5 cm. To calculate dr/dt, we need to know the rate at which the radius is changing. Unfortunately, that information is not provided in the question, so we cannot determine the rate of increase in the volume without that additional information.