Answer:
Step-by-step explanation:
Water is being pumped into an inflated rubber sphere at a constant rate of 0.03 cubic meters per second.
So, dV/dt = 0.03.
We want to show that dS/dt is directly proportional to 1/r.
In other words, we want to establish the relationship that dS/dt = k(1/r), where k is some constant.
First, the volume of a sphere V is given by:
Therefore:
Next, the surface area of a sphere S is given by:
Therefore:
We can divide both sides by 2:
We can substitute this into dV/dt. Rewriting:
So:
Since dV/dt = 0.03 or 3/100:
Therefore:
Where k = 3/50.
And we have shown that dS/dt is directly proportional to 1/r.