While building a snowman, a large snowball is created at the rate of 5 inches per hour. How fast is the volume of the snowball changing at the instant the snowball has a radius of 10 inches?
To determine the rate at which the volume of the snowball is changing, we need to find the derivative of the snowball's volume with respect to time.
The volume of a sphere can be calculated using the formula V = (4/3)πr^3, where V is the volume and r is the radius of the sphere.
Given that the snowball is being created at a rate of 5 inches per hour, we can express the rate of change of the radius with respect to time as dr/dt = 5 inches/hour.
To find the rate of change of the volume with respect to time, we differentiate the volume formula with respect to time:
dV/dt = d/dt [(4/3)πr^3]
Using the chain rule, we can express this as:
dV/dt = (4/3)π * (3r^2) * (dr/dt)
Now, we have the expression for the rate at which the volume is changing (dV/dt) in terms of the radius (r) and the rate of change of the radius with respect to time (dr/dt).
To find the rate at the instant the snowball has a radius of 10 inches, we substitute the given value into the expression:
dV/dt = (4/3)π * (3(10^2)) * (5)
Simplifying further:
dV/dt = (4/3)π * 300 * 5
dV/dt = 2000π
Hence, the volume of the snowball is changing at a rate of 2000π cubic inches per hour when the snowball has a radius of 10 inches.