Answer:
![\displaystyle \frac{dS}{dt}=\frac{3}{50r}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7BdS%7D%7Bdt%7D%3D%5Cfrac%7B3%7D%7B50r%7D)
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:
![\displaystyle V=\frac{4}{3}\pi r^3](https://tex.z-dn.net/?f=%5Cdisplaystyle%20V%3D%5Cfrac%7B4%7D%7B3%7D%5Cpi%20r%5E3)
Therefore:
![\displaystyle \frac{dV}{dt}=4\pi r^2\frac{dr}{dt}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7BdV%7D%7Bdt%7D%3D4%5Cpi%20r%5E2%5Cfrac%7Bdr%7D%7Bdt%7D)
Next, the surface area of a sphere S is given by:
![\displaystyle S=4\pi r^2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20S%3D4%5Cpi%20r%5E2)
Therefore:
![\displaystyle \frac{dS}{dt}=8\pi r\frac{dr}{dt}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7BdS%7D%7Bdt%7D%3D8%5Cpi%20r%5Cfrac%7Bdr%7D%7Bdt%7D)
We can divide both sides by 2:
![\displaystyle \frac{1}{2}\frac{dS}{dt}=4\pi r\frac{dr}{dt}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B2%7D%5Cfrac%7BdS%7D%7Bdt%7D%3D4%5Cpi%20r%5Cfrac%7Bdr%7D%7Bdt%7D)
We can substitute this into dV/dt. Rewriting:
![\displaystyle \frac{dV}{dt}=r\Big(4\pi r\frac{dr}{dt}\Big)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7BdV%7D%7Bdt%7D%3Dr%5CBig%284%5Cpi%20r%5Cfrac%7Bdr%7D%7Bdt%7D%5CBig%29)
So:
![\displaystyle \frac{dV}{dt}=\frac{1}{2}r\frac{dS}{dt}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7BdV%7D%7Bdt%7D%3D%5Cfrac%7B1%7D%7B2%7Dr%5Cfrac%7BdS%7D%7Bdt%7D)
Since dV/dt = 0.03 or 3/100:
![\displaystyle \frac{3}{100}=\frac{1}{2}r\frac{dS}{dt}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B3%7D%7B100%7D%3D%5Cfrac%7B1%7D%7B2%7Dr%5Cfrac%7BdS%7D%7Bdt%7D)
Therefore:
![\displaystyle \frac{dS}{dt}=\frac{3}{50r}=\frac{3}{50}\Big(\frac{1}{r}\Big)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7BdS%7D%7Bdt%7D%3D%5Cfrac%7B3%7D%7B50r%7D%3D%5Cfrac%7B3%7D%7B50%7D%5CBig%28%5Cfrac%7B1%7D%7Br%7D%5CBig%29)
Where k = 3/50.
And we have shown that dS/dt is directly proportional to 1/r.