What is the volume of a cylinder with a radius of 4 centimeters and a height of 5 centimeters when its radius is tripled?
Understand the Problem
The question is asking us to calculate the volume of a cylinder when its radius is tripled. We need to use the formula for the volume of a cylinder, V = πr^2h, and apply the given radius and height, then recalculate the volume with the tripled radius.
Answer
$1296\pi$ cubic inches
Answer for screen readers
$1296\pi$ cubic inches
Steps to Solve
- State the formula for the volume of a cylinder
The volume $V$ of a cylinder is given by the formula: $$ V = \pi r^2 h $$ where $r$ is the radius and $h$ is the height of the cylinder.
- Calculate the original volume
Given the radius $r = 4$ inches and the height $h = 9$ inches, the original volume $V_1$ is: $$ V_1 = \pi (4^2)(9) = \pi (16)(9) = 144\pi \text{ cubic inches} $$
- Triple the radius
The new radius $r_2$ is three times the original radius: $$ r_2 = 3r = 3(4) = 12 \text{ inches} $$
- Calculate the new volume
With the new radius $r_2 = 12$ inches and the same height $h = 9$ inches, the new volume $V_2$ is: $$ V_2 = \pi (12^2)(9) = \pi (144)(9) = 1296\pi \text{ cubic inches} $$
- Find the ratio of the new volume to the original volume or subtract the volumes.
The problem asks for the new volume with the tripled radius, not the difference.
The new volume is $1296\pi$ cubic inches.
$1296\pi$ cubic inches
More Information
The volume increases significantly when the radius is tripled because the radius is squared in the volume formula.
Tips
A common mistake is forgetting to square the radius when calculating the volume, or only multiplying the original volume by 3 instead of considering the squared relationship with the radius. Another mistake is calculating the difference in volumes when the problem only asks for the final volume.
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