Illustrative Mathematics Grade 8, Unit 5, Lesson 14: Finding Cylinder Dimensions

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Illustrative Math
Grade 8

Lesson 14: Finding Cylinder Dimensions

Let’s figure out the dimensions of cylinders.

Illustrative Math Unit 8.5, Lesson 14 (printable worksheets)

Lesson 14 Summary

In an earlier lesson we learned that the volume, V, of a cylinder with radius r and height h is

V = πr²h

We say that the volume depends on the radius and height, and if we know the radius and height, we can find the volume. It is also true that if we know the volume and one dimension (either radius or height), we can find the other dimension.

For example, imagine a cylinder that has a volume of 500π cm³ and a radius of 5 cm, but the height is unknown. From the volume formula we know that

500π = π · 25 · h

must be true. Looking at the structure of the equation, we can see that 500 = 25h. That means that the height has to be 20 cm, since 500 ÷ 25 = 20.

Now imagine another cylinder that also has a volume of 500π cm³ with an unknown radius and a height of 5 cm. Then we know that

500π = π · r² · 5

must be true. Looking at the structure of this equation, we can see that r² = 100. So the radius must be 10 cm.

Lesson 14.1 A Cylinder of Unknown Height

What is a possible volume for this cylinder if the diameter is 8 cm? Explain your reasoning.

• See Video for Whole Lesson

Lesson 14.2 What’s the Dimension?

The volume V of a cylinder with radius r is given by the formula V = πr²h.

1. The volume of this cylinder with radius 5 units is 50π cubic units. This statement is true:
   50π = 5²πh
   What does the height of this cylinder have to be? Explain how you know.
2. The volume of this cylinder with height 4 units is 36π cubic units. This statement is true
   36π = r²π4
   What does the radius of this cylinder have to be? Explain how you know.

Are you ready for more?

Suppose a cylinder has a volume of 36π cubic inches, but it is not the same cylinder as the one you found earlier in this activity.

1. What are some possibilities for the dimensions of the cylinder?
   • Show Answer

     Any combination of r²h = 36
     Example:
     r = 2, h = 9
     r = 3, h = 4

2. How many different cylinders can you find that have a volume of 36π cubic inches?
   • Show Answer

     There would be an infinite number of different cylinders since r and h need not be whole numbers.

Lesson 14.3 Cylinders with Unknown Dimensions

Each row of the table has information about a particular cylinder. Complete the table with the missing dimensions.

Lesson 14 Practice Problems

1. Complete the table with all of the missing information about three different cylinders.
2. A cylinder has volume 45π and radius 3. What is its height?
3. Three cylinders have a volume of 2826 cm³. Cylinder A has a height of 900 cm. Cylinder B has a height of 225 cm. Cylinder C has a height of 100 cm. Find the radius of each cylinder. Use 3.14 as an approximation for π.
4. A gas company’s delivery truck has a cylindrical tank that is 14 feet in diameter and 40 feet long.
   a. Sketch the tank, and mark the radius and the height.
   b. How much gas can fit in the tank?
5. Here is a graph that shows the water height of the ocean between September 22 and September 24, 2016 in Bodega Bay, CA.
   a. Estimate the water height at 12 p.m. on September 22.
   b. How many times was the water height 5 feet? Find two times when this happens.
   c. What was the lowest the water got during this time period? When does this occur?
   d. Does the water ever reach a height of 6 feet?

The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.

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