Question

Related rates problem?

a pint of ice cream is eaten causing the height of the ice cream in the cylindrical container to drop at a rate of 1 inch/min. the pint is 4 inches in diameter and 5.5 inches tall. how fast, in cubic inches per minute, is the volume of ice cream being eaten?

Answer 1

`22pi \ "in"^3"/min"`

Explanation:

First I want it to made apparently clear that we are finding the rate of volume or `(dV)/dt`.

We know from geometry that the volume of a cylinder is found by using the formula `V=pir^2h`.

Secondly, we know `pi` is a constant and our `h = 5.5` inches, `(dh)/(dt) = "1 inch/min"`.

Thirdly, our `r= 2` inches since `D=r/2` or `4/2`

We now find a derivative of our Volume using a Product Rule with respect to time, so:

`(dV)/dt=pi(2r(dr)/(dt)h+r^2(dh)/(dt))`

If we think about the cylinder, our radius isn't changing. That would mean the shape of the cylinder would have to change. Meaning `(dr)/(dt)=0`

so, by plugging in our varriable:

`(dV)/dt=pi(2(2)(0)(5.5)+2^2(5.5))` = `(dV)/dt=pi(2^2(5.5)) = 22pi`

with units `"inches"^3"/minute"`