1 answer

Related rates problem?

Question:

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?


Answers

#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"#

.

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