A state department of education, as a question on the public-release form of a statewide standardized test, released a question to the public in which students have to choose the value that most closely estimates the volume of a cylinder, given its radius, how tall it is (its height), and a value to use for π.
According to the problem, the cylinder is 6 inches tall and has a radius of 3 inches. We have a formula sheet that tells us the formula to use for the volume of a cylinder:
where r is the radius and h the height of a right cylinder. Plugging in the values, we get the answer of
Our answer choices are 36, 54, 113, and 170 cubic inches, of which 170 is the closest to our estimate of 169.56. I say “estimate” because 3.14 is not the exact value of π.
Although this problem seems straightforward (plug-n-chug with the formula), there are some things to watch out for. The biggest cause of error in these problems is when you forget that the radius and the height of the cylinder have to be given to you in the same units. Convert if necessary. Then, the computed volume will be in those cubic units.
Second, some textbooks make a big deal out of saying “the volume enclosed by the cylinder.” This is more mathematically precise, since a cylinder, by strict definition, is the shell of the figure and that shell has, in theory, a thickness of zero. These textbooks aren’t saying something wrong, but neither is your teacher who says “the volume of a cylinder.” One is more “precise” on a higher level of mathematical dialog, but we all know what both people mean. Don’t sweat the small stuff.
And finally, if you would like more practice on cylinder volume problems, visit our online library at VoxLearn.org and search for “cylinder volume.”
Try a few on your own
(1) Much to your surprise, your pet chocolate labrador dug a hole in your back yard that was in the shape of a perfect cylinder. The diameter was exactly 20 inches, in a perfect circle. The hole was 2 feet deep. How many gallons of water would it take to fill the hole to the top, assuming zero absorption of water by the dirt/clay in your back yard?
(2) How deep is a cylinder that has a volume of 48 cubic centimeters and a radius of 18 mm?
(3) The dolphin tank at Brookfield Zoo near Chicago holds an estimated 60 million liters of salt water. If the tank were a perfect cylinder, how tall and wide might it be? (There are many, many answers.)
The answers
(1) How many gallons in your dog’s hole?
First, we need to know the dimensions of a US gallon. It just happens to be exactly 231 cubic inches. Thus, if we can figure out how many cubic inches Rover dug out of your back yard, we divide by 231 to convert that to gallons. More on this later.
First we tackle the task of computing how many cubic inches are in this hole of yours. We are told the diameter is 20 inches, meaning the radius is 10 inches, since the radius is defined as half the diameter. And we have a height of 2 feet, or 24 inches.
Using the same formula we used in the main post for the volume of a right cylinder, we have
And to get the number of gallons, we divide our answer in cubic inches, 7,536, by 231. That gives us a final answer of about 32.6 gallons.
(2) Height of a cylinder with known radius and volume
For this one, we kind of have to “work backwards.” We are given a formula for volume given the radius and height, and here we need to compute the height given the radius and volume. So we solve the original formula for h, getting that variable on a side by itself:
In this problem, the volume is given as 48 cubic centimeters, and the radius is given as 18 millimeters. Well, that’s a little problem right there: the measurements are not in the same units. We should first convert both to the same units.
How many cubic millimeters are there in a cubic centimeter? Easy. Since a centimeter is 10 millimeters, there are 10-cubed, or 1,000, cubic millimeters in a cubic centimeter.
So we can write our volume in terms of cubic millimeters by multiplying by 1,000, giving us a volume of 48,000 cubic millimeters. The radius is 18 millimeters (same units as volume, but not cubed), so
(3) The water, the dolphins, and the tank
Before we get into the problem, here is a lovely photograph, in black and white, taken by Cassie K. Tait off the coast of New Zealand.
I only brought that photo in because I wanted to show you what a happy dolphin looks like, jumping around in the water, endless gallons of it, in the warm climate. Unfortunately, in our zoo problem, we are limited to 60 million liters, which is not much compared to the dolphin in the picture.
In terms of measurement, we know that there are 1,000 liters in a cubic meter. Thus, we will use meters in the calculations of width and height for our problem. Furthermore, we can eliminate a few zeroes, since we know that 60 million liters comes out to 60,000 cubic meters.
The task at hand is to determine what combination(s) of height and width of a right cylinder will give us a volume of 60,000 cubic meters. We’ll use 3.14 for π. Mathematically speaking,
We can see that as we decrease the radius, the height must go up, and vice versa. That makes sense, since the water we’re taking away by making the tank smaller on the surface, we have to add in the depth. Let’s look at a few values to demonstrate this principle.
We compute the height just like we did in #2: If V = π r2 h, then
Of course, we have to keep in mind that some of these values might get a little ridiculous, since the dolphin would need room to swim around (and probably even jump), even if he were in a tank at the zoo. This means that depths of less than about 5 meters would probably be unacceptable to the zoo owners. Even though the tank would still hold 60 million liters, in other words, we still have to check that our values make sense in the context of the problem.
Illinois Alignment
Grade Level: 8
Illinois Assessment Framework: 7.8.04 (8th grade) Solve problems involving the volume or surface area of a right rectangular prism, right circular cylinder, or composite shape using an appropriate formula or strategy.
Illinois Learning Standard: 7.C.3b (middle school) Use concrete and graphic models and appropriate formulas to find perimeters, areas, surface areas and volumes of two- and three-dimensional regions.