Well, well, well, looks like we have ourselves a hot and steamy glass sphere on our hands! Let's dive into the fiery world of rate of change, shall we?
To start things off, we can use the formula for the volume of a sphere: V = (4/3)πr³. Now we need to figure out how the volume changes with respect to time.
Since the radius is in direct proportion to the temperature, we can say that r = kT, where k is just some constant of proportionality. (I know, it sounds like a bad pick-up line for a glass sphere, but bear with me here.)
Now, let's differentiate both sides of this equation with respect to time (t). Using the chain rule, we get:
(d/dt) r = (d/dt) (kT)
The left side represents the rate of change of the radius, which we'll call dr/dt. And since the temperature (T) doesn't change with time, the right side becomes:
dr/dt = k (d/dt) T
But we're not looking for the rate of change of the radius - we want the rate of change of the volume, which is V. So let's differentiate the volume equation with respect to time (t):
(d/dt) V = (d/dt) [(4/3)πr³]
Using the chain rule again, we get:
(d/dt) V = (4/3)π (d/dt) (r³)
However, we know that dr/dt = k (d/dt) T from before. So we can rewrite the right side as:
(d/dt) V = (4/3)π (d/dt) (k³T³)
Applying the power rule and multiplying through, we get:
(d/dt) V = 4πk³ (d/dt) T³
And remember, (d/dt) T³ is just the rate of change of the temperature with respect to time. So, in the end, we have our expression for the rate of change of the volume in terms of r and T:
(d/dt) V = 4πk³ (d/dt) T³
Now you can go impress your friends with your knowledge of hot glass spheres and their changing volumes. Just make sure you don't get burned in the process!