# rate of growth of a leaking oil tank

• Mar 30th 2009, 07:22 AM
allywallyrus
rate of growth of a leaking oil tank
A gas tank on a dock has a small puncture and is leaking gas at the rate of $\displaystyle 1cm^3/min$ into a lake. It forms a circular slick that is 1mm thick on the surface of the water.

so as a function of time the amount of gas leaked is $\displaystyle V=1cm^3 * m$ but i also have to state the radius as a function of volume and as a function of time. im supposing i should approach the radius as volume as a very short cylinder, which i think i would be $\displaystyle r=\sqrt{\frac{\pi*h}{V}}$ and as a function of a time as $\displaystyle r=\sqrt{\frac{\pi*h}{1cm^3 * min}}$
this is my first time attempting a question like this and im not even sure its in the correct forum so any advice would be appreciated :)
• Mar 30th 2009, 10:38 AM
HallsofIvy
Quote:

Originally Posted by allywallyrus
A gas tank on a dock has a small puncture and is leaking gas at the rate of $\displaystyle 1cm^3/min$ into a lake. It forms a circular slick that is 1mm thick on the surface of the water.

so as a function of time the amount of gas leaked is $\displaystyle V=1cm^3 * m$ but i also have to state the radius as a function of volume and as a function of time. im supposing i should approach the radius as volume as a very short cylinder, which i think i would be $\displaystyle r=\sqrt{\frac{\pi*h}{V}}$ and as a function of a time as $\displaystyle r=\sqrt{\frac{\pi*h}{1cm^3 * min}}$

For a cylinder of height h and radius r, $\displaystyle V= \pi r^2h$, then $\displaystyle r^2= \frac{V}{\pi h}$ so $\displaystyle r= \sqrt{\frac{V}{\pi h}}$. It looks to me like you have the fraction upside-down.

Quote:

this is my first time attempting a question like this and im not even sure its in the correct forum so any advice would be appreciated :)
To find the rate at which r is increasing, differentiate $\displaystyle \frac{dr}{dt}= \sqrt{\frac{1}{\pi h}}V^{1/2}$ with respect to t. h is constant, h= 0.1 cm, and dV/dt= 1.
• Mar 30th 2009, 10:47 AM
allywallyrus
ah, so i wasnt that far off at least. thanks :)
also the question wasnt asking the rate that r was increasing, but just how to determine the radius after so much time had elapsed, so would i also need that formula?