Answer:
The rate at which the volume is decreasing when the snowball has a diameter of 4 inches is approximately 251.327 cubic inches per minute.
Step-by-step explanation:
Let suppose that snowball geometry correspond to a sphere. In this case, the volume of the sphere is:
[tex]V = \frac{4\pi}{3}\cdot R^{3}[/tex] (Eq. 1)
Where:
[tex]R[/tex] - Radius of the sphere, measured in inches.
[tex]V[/tex] - Volume of the sphere, measured in cubic inches.
We know that volume decreases due to melting, so we need an expression of the rate of change of volume in time. By differentiating (Eq. 1) we get:
[tex]\frac{dV}{dt} = 4\pi\cdot R^{2}\cdot \frac{dr}{dt}[/tex] (Eq. 2)
Where:
[tex]\frac{dV}{dt}[/tex] - Rate of change of the volume of the snowball in time, measured in cubic inches per minute.
[tex]\frac{dr}{dt}[/tex] - Rate of change of radius of the snowball in time, measured in inches per minute.
If we know that [tex]R = 2\,in[/tex] and [tex]\frac{dr}{dt} = -5\,\frac{in}{min}[/tex], the rate at which the volume of the snowball is decreasing is:
[tex]\frac{dV}{dt} = 4\pi\cdot (2\,in)^{2}\cdot \left(-5\,\frac{in}{min} \right)[/tex]
[tex]\frac{dV}{dt}\approx -251.327\,\frac{in^{3}}{min}[/tex]
The rate at which the volume is decreasing when the snowball has a diameter of 4 inches is approximately 251.327 cubic inches per minute.
