Question

if the height of a cylinder is doubled, what happens to the volume?
a. it remains the same
b. it doubles
c. it quadruples
d. it halves

what is the formula for the volume of a sphere?
a. \\( \pi r ^ { 2 } h \\)
b. \\( \frac { 3 } { 4 } \pi r ^ { 3 } \\)
c. \\( 4 \pi r ^ { 3 } \\)
d. \\( \frac { 4 } { 3 } \pi r ^ { 3 } \\)

Explanation:

Step1: Recall the volume formula 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: Analyze the effect of doubling the height

Let the original height be \(h_{1}=h\) and the new height be \(h_{2} = 2h\). The original volume \(V_{1}=\pi r^{2}h_{1}=\pi r^{2}h\), and the new volume \(V_{2}=\pi r^{2}h_{2}=\pi r^{2}(2h)=2(\pi r^{2}h) = 2V_{1}\).

Step3: Recall the volume formula of a sphere

The volume formula of a sphere is \(V=\frac{4}{3}\pi r^{3}\), where \(r\) is the radius of the sphere.

Answer:

For the first question: B. It doubles.
For the second question: D. \(\frac{4}{3}\pi r^{3}\)