Reasoning A cylinder with radius 5 ft and height 8 ft has its radius tripled. How many times greater is the volume of the larger cylinder than the smaller cylinder? Use pencil and paper. Explain how the volume of the cylinder changes if its height were also tripled
To find how many times greater the volume of the larger cylinder is than the smaller cylinder, we need to compare their volumes.
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.
For the smaller cylinder with radius 5 ft and height 8 ft, the volume is V1 = π(5^2)(8) = 200π ft^3.
If the radius is tripled, then the new radius would be 3 * 5 = 15 ft.
The volume of the larger cylinder with radius 15 ft and height 8 ft is V2 = π(15^2)(8) = 1800π ft^3.
To find how many times greater the volume of the larger cylinder is than the smaller cylinder, we can divide V2 by V1:
V2/V1 = (1800π ft^3) / (200π ft^3)
Simplifying, π cancels out:
V2/V1 = 1800/200
V2/V1 = 9
Therefore, the volume of the larger cylinder is 9 times greater than the smaller cylinder.
If the height of the cylinder is also tripled, the volume changes.
Let's say the original height of the cylinder is h1. If it is tripled, the new height would be 3 * h1.
The volume of the original cylinder with radius 5 ft and height h1 is V1 = π(5^2)(h1) = 25πh1 ft^3.
If the height is tripled, the volume of the cylinder becomes V2 = π(5^2)(3h1) = 75πh1 ft^3.
Comparing the new volume V2 to the original volume V1:
V2/V1 = (75πh1 ft^3) / (25πh1 ft^3)
Simplifying, π and h1 cancel out:
V2/V1 = 75/25
V2/V1 = 3
Therefore, if the height of the cylinder is also tripled, the volume becomes 3 times greater than the original volume.