Final answer:
To calculate the rate of change of the volume of a sphere with a given volume and a radius increasing at a constant rate, we use the derivative of the volume formula with respect to time. After finding the radius using the given volume, we substitute it and the rate of radius change into the differentiated formula to find the answer, which is 176.000 cubic feet per second.
Step-by-step explanation:
To find the rate of change of the volume of a sphere, we need to use the formula for the volume of a sphere: V = (4/3)πr^3
First, we differentiate the volume formula with respect to time (t) to find the rate of change of volume.
dV/dt = 4πr^2(dr/dt)
Given that dr/dt is the rate of change of the radius, which is 7 feet per second, we substitute this value and the radius corresponding to the volume 274,274 cubic feet.
To find the radius from the given volume, we can rearrange the volume formula to solve for r: r = √[3V/(4π)]^(1/3)
After calculation, we substitute the radius back into the differentiated formula to get the rate of change of the volume:
dV/dt = 4πr^2(7)
The result is the rate at which the volume is increasing at the moment when the volume is 274,274 cubic feet. The correct answer from the available options is 176.000 cubic feet per second.