BAre the radii the same? Are the heights the same?
CUse the expressions you wrote in Part A of the exercise.
DCompare the volumes of Cylinders B, C, and D with the volume of Cylinder A. How do the volumes change?
A
Radius (cm)
Height (cm)
Volume (cm^3)
Cylinder A
1
1
Ï€(1)^1(1)
Cylinder B
1
2
Ï€(1)^1(2)
Cylinder C
2
1
Ï€(2)^1(1)
Cylinder D
2
2
Ï€(2)^1(2)
BThe height of Cylinder B is twice the height of Cylinder A. The radius of Cylinder C is twice the radius of Cylinder A. The radius and height of Cylinder D are twice the radius and height of Cylinder A.
C
Radius (cm)
Height (cm)
Volume (cm^3)
Cylinder A
1
1
3.14
Cylinder B
1
2
6.28
Cylinder C
2
1
12.57
Cylinder D
2
2
25.13
DSee solution.
Practice makes perfect
In the table we can see the dimensions for four cylinders.
Radius (cm)
Height (cm)
Volume (cm^3)
Cylinder A
1
1
Cylinder B
1
2
Cylinder C
2
1
Cylinder B
2
2
Our goal is to write an equation representing the volume of each cylinder. For that, let's remember the formula for the volume of a cylinder. Here r is the radius of the circular base, and h is the height of the cylinder.
V=Ï€ r^2h
To write the equations, we should substitute the radius and the height of each of the cylinders into the formula. Let's do it then!
Radius (cm)
Height (cm)
Volume (cm^3)
Cylinder A
1
1
Ï€( 1)^2( 1)
Cylinder B
1
2
Ï€( 1)^2( 2)
Cylinder C
2
1
Ï€( 2)^2( 1)
Cylinder B
2
2
Ï€( 2)^2( 2)
We are asked to compare the dimensions of Cylinder A with the dimensions of Cylinders B, C, and D. Let's take a look at the table one more time.
Radius (cm)
Height (cm)
Volume (cm^3)
Cylinder A
1
1
Cylinder B
1
2
Cylinder C
2
1
Cylinder D
2
2
First, we see that Cylinders A and B have the same radius. Also, the height of Cylinder B is twice the height of Cylinder A.
Radius (cm)
Height (cm)
Volume (cm^3)
Cylinder A
1
1
Cylinder B
1
2 = 2( 1)
Cylinder C
2
1
Cylinder D
2
2
Now, Cylinders A and C have the same height, but the radius of Cylinder C is twice the radius of Cylinder A.
Radius (cm)
Height (cm)
Volume (cm^3)
Cylinder A
1
1
Cylinder B
1
1
Cylinder C
2 = 2( 1)
1
Cylinder D
2
2
Last, the radius and height of Cylinder D are twice the radius and height of Cylinder A.
Radius (cm)
Height (cm)
Volume (cm^3)
Cylinder A
1
1
Cylinder B
1
1
Cylinder C
2
1
Cylinder D
2 = 2( 1)
2 = 2( 1)
We are asked to complete the table. Notice that in Part A of the exercise we wrote the expressions representing the volumes of each cylinder.
Radius (cm)
Height (cm)
Volume (cm^3)
Cylinder A
1
1
Ï€ (1)^2(1)
Cylinder B
1
2
Ï€ (1)^2(2)
Cylinder C
2
1
Ï€ (2)^2(1)
Cylinder D
2
2
Ï€ (2)^2(2)
This means we should only evaluate the expressions. Let's do it then! We can start with the first expression.
The volume of the second cylinder is about 6.28 cm^3. Following similar steps as before we can find the other two volumes. Here are the results.
Radius (cm)
Height (cm)
Volume (cm^3)
Cylinder A
1
1
π (1)^2(1) ≈ 3.14
Cylinder B
1
2
π (1)^2(2) ≈ 6.28
Cylinder C
2
1
π (2)^2(1) ≈ 12.57
Cylinder D
2
2
π (2)^2(2)≈ 25.13
We are asked to describe how changing the dimensions of a cylinder affects the cylinder's volume. Let's then take a look at the completed table from Part C of the exercise.
Radius (cm)
Height (cm)
Volume (cm^3)
Cylinder A
1
1
3.14
Cylinder B
1
2
6.28
Cylinder C
2
1
12.56
Cylinder D
2
2
25.12
For now, let's focus on Cylinders A and B. We said that the height of Cylinder B is twice the height of Cylinder A. Notice that the same is true for their volumes.
Radius (cm)
Height (cm)
Volume (cm^3)
Cylinder A
1
1
3.14
Cylinder B
1
2
6.28=2( 3.14)
Cylinder C
2
1
12.56
Cylinder D
2
2
25.12
This means that when the height is doubled, the volume is twice the original volume. Let's now compare the volumes of of Cylinders A and C.
Radius (cm)
Height (cm)
Volume (cm^3)
Cylinder A
1
1
3.14
Cylinder B
1
2
6.28
Cylinder C
2
1
12.56=4( 3.14)
Cylinder D
2
2
25.12
This time the radius was doubled, which made the volume four times the original volume. Let's now shift our focus to the last cylinder and its volume.
Radius (cm)
Height (cm)
Volume (cm^3)
Cylinder A
1
1
3.14
Cylinder B
1
2
6.28
Cylinder C
2
1
12.56
Cylinder D
2
2
25.12=8( 3.14)
The radius and height of Cylinder D double the radius and height of Cylinder A. As a result, the volume is eight times the volume of Cylinder A.
