Question

1. the diameter of a cylinder is doubled, but its height remains the same. how does its volume change?

a. it doubles
b. it remains the same
c. it quadruples
d. it triples

Explanation:

Step1: Write the formula for the volume of a cylinder

The volume formula of a cylinder is \(V = \pi r^{2}h\), where \(r\) is the radius and \(h\) is the height.

Step2: Express the relationship between diameter and radius

Since \(d = 2r\) (diameter \(d\) and radius \(r\)), if the diameter is doubled (\(d_{new}=2d\)), then the new radius \(r_{new}=\frac{d_{new}}{2}=\frac{2d}{2}= 2r\) (because \(d = 2r\) originally).

Step3: Calculate the new volume

Substitute \(r_{new}=2r\) into the volume formula. The new volume \(V_{new}=\pi(2r)^{2}h\). Using the formula \((ab)^{n}=a^{n}b^{n}\), we have \(V_{new}=\pi\times4r^{2}h = 4\pi r^{2}h\).

Step4: Compare the new volume with the original volume

Since the original volume \(V=\pi r^{2}h\), and \(V_{new} = 4V\). So when the diameter (and thus the radius) is doubled while the height remains the same, the volume of the cylinder quadruples.

Answer:

C. It quadruples