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Chapter 1: Problem 10

By how much does the volume of a cylinder change if the radius is halved and the height is doubled? a) The volume is quartered. d) The volume doubles. b) The volume is cut in half. e) The volume c) There is no change in the volume. quadruples.

Short Answer

Expert verified
Answer: b) The volume is cut in half.

Step by step solution

01

Write down the volume formula for a cylinder

The formula for the volume of a cylinder is given by: \[\text{Volume} = \pi r^2h\] where \(r\) is the radius of the cylinder, and \(h\) is the height.
02

Examine the change in the radius and height

The problem states that the radius is halved, and the height is doubled. Let's represent the initial radius as \(r\) and the height as \(h\). Then, the new radius becomes \(\frac{1}{2}r\), and the new height is \(2h\).
03

Calculate the new volume with the changed radius and height

Substitute the new radius and height into the volume formula: \[\text{New Volume} = \pi(\frac{1}{2}r)^2(2h)\]
04

Simplify the new volume equation

Simplify the equation from Step 3: \[\text{New Volume} = \pi \left(\frac{1}{4}r^2\right)(2h)\]
05

Compare the initial volume with the new volume

Compare the initial and new volume equations: \[\frac{\text{New Volume}}{\text{Initial Volume}} = \frac{\pi\left(\frac{1}{4}r^2\right)(2h)}{\pi r^2h}\]
06

Simplify the ratio

Simplify the ratio between the initial and new volumes: \[\frac{\text{New Volume}}{\text{Initial Volume}} = \frac{\left(\frac{1}{4}r^2\right)(2h)}{r^2h}\] The \(r^2\) and \(h\) terms cancel each other out, and we are left with: \[\frac{\text{New Volume}}{\text{Initial Volume}} = \frac{1}{2}\] So the new volume is half of the initial volume.
07

Choose the correct answer

From the given options, the correct answer is: b) The volume is cut in half.

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