Question

Two cylinders are given. A small cylinder has a radius of 2 meters, and a larger cylinder has a radius of 7 meters. The larger cylinder is a dilation of the smaller cylinder.

Part a: Determine the scale factor. Show or explain all the necessary steps.

A) 2.5
B) 3.5
C) 4.5
D) 5.5
E) 6.5
Part b: If the volume of the smaller cylinder is 24π ³, show how the scale factor can be used to determine the volume of the larger cylinder.
A) 56πm³
B) 84πm³
C) 96πm³
D) 112πm³
E) 126πm³

Answer 1

The scale factor between the two cylinders is 3.5. To determine the volume of the larger cylinder, you multiply the volume of the smaller cylinder by the cube of the scale factor.

Step-by-step explanation:

Part a:

To find the scale factor between the two cylinders, divide the radius of the larger cylinder by the radius of the smaller cylinder. The scale factor is equal to the ratio of the radii.

Scale factor = Radius of larger cylinder / Radius of smaller cylinder = 7 m / 2 m = 3.5

Therefore, the scale factor is 3.5.

Part b:

To determine the volume of the larger cylinder, multiply the volume of the smaller cylinder by the cube of the scale factor.

Volume of larger cylinder = Volume of smaller cylinder x (Scale factor)³

Volume of larger cylinder = 24π x (3.5)³ = 24π x 42.875 = 1029π

Therefore, the volume of the larger cylinder is 1029πm³.

Answer 2

if the volume of the smaller cylinder is \
`(24πm^3\)`, the volume of the larger cylinder can be calculated as
`\(24πm^3 * 3.5^3 = 112πm^3\).`

Part a: B) 3.5

Part b: D) 112πm³

Step-by-step explanation:

The scale factor is determined by comparing the radii of the smaller and larger cylinders. Given the smaller cylinder has a radius of 2 meters and the larger cylinder has a radius of 7 meters, the scale factor can be calculated as the ratio of their radii:
`\( (7)/(2) = 3.5 \).` Thus, the scale factor between the two cylinders is 3.5.

To find the volume of the larger cylinder based on the scale factor, we know that the volume of a cylinder is directly proportional to the cube of its radius. As the scale factor is 3.5, which represents the ratio of the radii, the volume of the larger cylinder compared to the smaller one will be \(3.5^3\) times greater (since volume is proportional to the cube of the scale factor). Therefore, if the volume of the smaller cylinder is \
`(24πm^3\)`, the volume of the larger cylinder can be calculated as
`\(24πm^3 * 3.5^3 = 112πm^3\).`