What is the rate of this oil slick spill? Cyclinder

[simongagne53]

Hi everyone! Thanks in advance for your time

 

[galactus]

An oil spill is being cleaned up by deploying bacteria that eat up the oil at 4 cubic feet per hour. The oil spill itself is modeled in the form of a very thin cylinder whose height is the thickness of the oil slick. When the thickness of the slick is 0.001 feet, the cylinder is 500 ft in diameter. If the height is decreasing at 0.0005 feet per hour, at what rate is the area of the slick changing

The slick is a cylinder with a tiny height.

Volume is given by \(\displaystyle V={\pi}r^{2}\) and area by \(\displaystyle A={\pi}r^{2}\).

\(\displaystyle \frac{dV}{dt}={\pi}\left(r^{2}\frac{dh}{dt}+h\cdot 2r\frac{dr}{dt}\right)\)..........[1]

\(\displaystyle \frac{dA}{dt}=2{\pi}r\cdot\frac{dr}{dt}\)..............[2]

Now, we are given \(\displaystyle \frac{dV}{dt}=-4, \;\ h=.001, \;\ \frac{dh}{dt}=-.0005, \;\ r=250\).

We need \(\displaystyle \frac{dA}{dt}\). To find this we need dr/dt. It can be found by subbing in all the knowns into [1] and solving for dr/dt.

Then, sub the just found dr/dt into [2] and find dA/dt as needed.