1. ## modelling functions

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

a) Find the amount of gas leaked as a function of time.

b) State the radius of the slick as a function of its volume

c) State the radius of the slick as a function of time.

d) What is the radius of the slick after 40 minutes?

2. Hi

You say nothing about how much gas had already leaked out. Or is this assumed to be zero ?

You have that gas is leaking out with $\displaystyle 1 cm^{3}/min$ , so lets say the time variable $\displaystyle t$ is in minutes.

Call $\displaystyle y$ the amount of gas leaked out. Then:

$\displaystyle \frac{dy}{dt}=-1\cdot 10^{-6} m^{3}/min \; \Rightarrow y(t) = -t\cdot 10^{-6} + C \; m^{3}$ Set $\displaystyle C = 0$ if none had already out.

The gas forms a cylinder on the water.

$\displaystyle V = \pi r^{2} h \;$ This gives us: $\displaystyle r = \sqrt{\frac{1000V}{\pi}}$ , since we are using SI-units here, that is calling 1mm = 0.001m .

To find the radius as a function of the amount of gas, interchange V for $\displaystyle y(t)$