## To understand the relationship between dV/dt and dr/dt, let's break down the problem step-by-step and explain each component.

First, let's consider the formula for the volume of a sphere: V = (4/3) Ï€ r^3. Here, V represents the volume, and r represents the radius of the sphere.

Now, we are interested in how the volume changes over time, or in other words, the derivative of V with respect to time, which is represented by dV/dt. This tells us how fast the volume of the sphere is changing.

To express dV/dt in terms of dr/dt, we need to find a relationship between the rate at which the volume changes (dV/dt) and the rate at which the radius changes (dr/dt).

To do this, we can differentiate the volume formula with respect to time using the chain rule. The derivative of V with respect to time (dV/dt) is equal to the derivative of V with respect to r (dV/dr) multiplied by the derivative of r with respect to time (dr/dt).

The derivative of V with respect to r is found by differentiating the volume formula with respect to r, which gives us dV/dr = 4Ï€r^2. This represents the rate at which the volume changes concerning the radius.

Finally, we substitute dV/dr = 4Ï€r^2 and dr/dt into the equation to find the relationship between dV/dt and dr/dt:

dV/dt = (4Ï€r^2) * (dr/dt)

This equation shows that the rate of change of the volume (dV/dt) is equal to the rate of change of the radius (dr/dt) multiplied by a constant value of 4Ï€r^2.

In words, the relationship can be described as follows: "The rate at which the volume of the sphere is changing (dV/dt) equals the rate at which the radius is changing (dr/dt) multiplied by 4Ï€r^2." This means that as the radius changes, the volume changes at a rate proportional to the current radius multiplied by a constant value of 4Ï€r^2.