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?

1 Answer
Jun 1, 2018

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"