A roll of paper towels has the given dimensions. Determine, to the nearest hundredth, the volume of the roll.
This roll of paper towels is in the shape of a cylinder. Now a roll of paper towels doesn’t have paper towels to fill up the entire cylinder. The center of it is hollow. That’s the holder of the paper towels. So in order to find the volume, we need to find the volume of the large cylinder, the whole thing, and take away the small cylinder on the inside that’s hollow, that doesn’t have paper towels.
The volume 𝑉 of a cylinder is equal to the area of the base 𝐵 times the height h. But since the base of a cylinder is a circle, we can replace 𝐵 with 𝜋𝑟 squared, where 𝑟 is the radius. So to find the volume of the paper towels, as we said before, we need to take the volume of the large cylinder and subtract the volume of the small cylinder. Therefore, we need the radii of each cylinder and the height of each cylinder. Let’s begin with the large cylinder.
The height of the large cylinder is thirty centimeters. However, we’re not given the radius. The sixteen centimeters is a diameter. That’s the complete width of a circle, from one end of the circle to the other. The radius is from the centre to the outside of a circle. It’s exactly half of a diameter. So we need to take sixteen divided by two. So our radius is eight centimeters.
Now looking at our smaller cylinder, our height is still thirty, and our radius we need to find because they give us the diameter of the smaller cylinder. So we need the radius. So we need to take four and divide it by two. This means our radius is two centimeters.
Now we need to simplify. First let’s square the eight and square the two. Now we need to multiply sixty-four and thirty together, and then we also need to multiply four and thirty together. Now we subtract one thousand nine hundred and twenty 𝜋 minus one hundred and twenty 𝜋, resulting in one thousand eight hundred 𝜋.
Now it says determine to the nearest hundredth. That means we need to multiply one thousand eight hundred by 𝜋 and then round two decimal places. We get five thousand six hundred and fifty-four point eight six six seven seven seven. Now to round two decimal places, we need to decide if the six should stay a six or if it should round up to a seven. So we look at the number to the right of it. Since it is a six, which is five or larger, we will round our six, the first six, up to a seven.
Therefore, the volume of the roll of the paper towels is five thousand six hundred and fifty-four point eight seven centimeters cubed.