Cylinder problem

Question

Hi, my question is this: I rolled up piece of paper that measures 8.5 inches x 11 inches two different ways to create 2 different cylinders that hold a different amount, this shows that the two cylinders have different volumes, I need to find how much more space is inside the shorter cylinder compared to the taller one.

Answer 1

Hi Jackson.

First, remember that the "space inside" a cylinder is called Volume. Recall that the volume of a cylinder is found with the formula V=πr²h, where r=radius of the top or bottom circle (both being the same), and h=height of the cylinder (distance between the top & bottom).

Your two cylinders will have the following measurements:

C₁ has a circumference of 8.5" and a height of 11";

C₂ has a circumference of 11" and a height of 8.5".

But for our formula, we need the radius, not the circumference. Recall again, however, that C=2πr; which can be rewritten as r=C/(2π). So we can substitute this in for r in the volume formula:

V=π(r)²h => π(C/(2π))²h => (C²/(4π))h.

After that, all you need to do is "plug and play", or plug the values you have in the correct places in your formula, to find each cylinder's volume; then subtract.

Answer 2

When you ask how much a cylinder will hold, it is asking what the volume is for the interior of the cylinder formed. Area of the cross section times the length of the cylinder.

If you take the 8.5" x 11" paper and roll it to make a 1" diameter cylinder lengthwise (11") it will have a volume of Pi x r^2 x 11" . If you take the same paper and roll it along the width (8.5") the volume will be PI x r^2 x 8.5".

-Let's look at the volume for rolling the paper lengthwise. Pi=3.14, r =0.5" and the length is 11"

3.14 x 0.5^2 x 11 = 3.14 x 0.25 x 11= 8.64 in^3

-Let's look at the volume for rolling the paper along the width. PI=3.14, r=0.5" and the length is 8.5".

3.14 x 0.5^2 in^2 x 8.5 in = 3.14 x 0.25 in^2 x 8.5 in = 6.68 in^3

Volumes are 8.64 cubic inches and 6.68 cubic inches if rolled lengthwise, verses along the width, respectively, 8.64/6.68 = 1.29. There is 29% more space (volume) when rolled lengthwise.