# A page measuring x by y is curved around to make an open cylinder with height y. Find the volume of the cylinder in terms of x and y.   Ans: `V=(x^2 y)/(4pi)`

Expert Answers
lemjay | Certified Educator

To determine the volume, apply the formula:

`V= pir^2h`

Since the height of the cylinder formed is y, plug-in h=y to the formula.

`V=pir^2y`

To determine the r, refer to the figure below. Notice that the side x in the rectangle is the perimeter of the base of the cylinder.

Since the base of a cylinder is a circle, apply the formula:

`P=2pir`

Then, plug-in P=x.

`x=2pir`

And divide both sides by 2pi.

`x/(2pi)=(2pir)/(2pi)`

`x/(2pi)=r`

Now that r is expressed in terms of x, plug-in this to V.

`V=pir^2y`

`V=pi(x/(2pi))^2y`

`V=pi*x^2/(4pi^2)*y`

`V=(pi*x^2*y)/(4pi^2)`

`V=(x^2y)/(4pi)`

Hence, the volume of the cylinder formed is `V=(x^2y)/(4pi)` .

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