What is the volume of a cylinder with radius 4 and height 5 if the radius is tripled?
Understand the Problem
The question asks to calculate the volume of a cylinder, given its radius and height, and then recalculate the volume when the radius is tripled. We need to use the formula for the volume of a cylinder, which is V = πr²h, where V is the volume, r is the radius, and h is the height.
Answer
Original volume: $90\pi$ cubic inches. New volume: $810\pi$ cubic inches.
Answer for screen readers
The original volume is $90\pi$ cubic inches, and the new volume with the tripled radius is $810\pi$ cubic inches.
Steps to Solve
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Calculate the initial volume. We will use the formula for the volume of a cylinder: $V = \pi r^2 h$. Given $r = 3$ inches and $h = 10$ inches, we substitute these values into the formula:
$V = \pi (3^2)(10)$
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Simplify to find the initial volume. Calculate the square of the radius and then multiply by the height and $\pi$:
$V = \pi (9)(10) = 90\pi$ cubic inches.
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Triple the radius. The new radius will be three times the original radius:
$r_{new} = 3 \times 3 = 9$ inches.
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Calculate the new volume with the tripled radius. Using the same formula $V = \pi r^2 h$, substitute the new radius $r_{new} = 9$ and the original height $h = 10$:
$V_{new} = \pi (9^2)(10)$
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Simplify to find the new volume. Calculate the square of the new radius and then multiply by the height and $\pi$:
$V_{new} = \pi (81)(10) = 810\pi$ cubic inches.
The original volume is $90\pi$ cubic inches, and the new volume with the tripled radius is $810\pi$ 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. Another mistake is to only multiply the initial volume by 3 instead of $3^2 = 9$ when the radius is tripled.
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