Aspherical balloon filled with gas has a leak that causes the gas to escape at a rate of 1.5 m3 /min. at what rate is the surface area of the balloon shrinking when the radius is 4 cm?

The volume of the balloon is a function of the radius

So since the volume is changing in time, both volume and radius are functions of time

Now take the derivative of both sides with respect to t, and not forgetting the chain rule

and now we use dV/dt=1.5 and solve for dr/dt

since we want the rate of surface area change when r=0.04m, we can put that value in for r


Ok now look at the surface area formula:


Now since we want its rate of change, we take the derivative again remembering to use the chain rule



Now we use the value of dr/dt we found earlier and put in 0.04 for r

Alinear relationship would look like a straight line on the graph. it could have a positive negative or 0 slope. so it could be a line potentially going in any direction but it must be perfectly straight.