Question

1. the volume of a cylinder is given by ( v = pi r^2 h ). if the radius is doubled and the height is halved, how does the volume change? a) the volume is halved. b) the volume is unchanged. c) the volume is doubled. d) the volume is quadrupled (multiplied by a factor of 4).

Explanation:

Step1: Define new radius and height

Let original radius be \( r \), original height be \( h \). New radius \( r' = 2r \), new height \( h'=\frac{h}{2} \).

Step2: Calculate new volume

New volume \( V'=\pi (r')^{2}h' \). Substitute \( r' = 2r \), \( h'=\frac{h}{2} \):
\( V'=\pi(2r)^{2}\cdot\frac{h}{2}=\pi\cdot4r^{2}\cdot\frac{h}{2}=2\pi r^{2}h \).
Original volume \( V = \pi r^{2}h \), so \( V' = 2V \)? Wait, no, wait: Wait, \( (2r)^2 = 4r^2 \), times \( \frac{h}{2} \): \( 4r^2 \times \frac{h}{2}=2r^2h \), times \( \pi \): \( 2\pi r^2h \). Wait, original \( V=\pi r^2h \), so \( V' = 2V \)? Wait, no, I made a mistake. Wait, \( (2r)^2 = 4r^2 \), then multiply by \( \frac{h}{2} \): \( 4r^2 \times \frac{h}{2} = 2r^2h \), so \( V' = \pi \times 2r^2h = 2\pi r^2h = 2V \)? Wait, no, wait the options: C is volume doubled. Wait, let's recheck:

Original \( V = \pi r^2 h \).

New \( r = 2r \), new \( h = h/2 \).

So \( V' = \pi (2r)^2 (h/2) = \pi \times 4r^2 \times h/2 = \pi \times 2r^2 h = 2 \times (\pi r^2 h) = 2V \). So the volume is doubled. Wait, but wait, maybe I miscalculated? Wait, \( (2r)^2 = 4r^2 \), times \( h/2 \): 4*(1/2)=2, so yes, 2 times original volume. So the volume is doubled. So option C.

Wait, but let me check again. Original volume: \( V = \pi r^2 h \). New radius: 2r, new height: h/2. So new volume \( V' = \pi (2r)^2 (h/2) = \pi \times 4r^2 \times h/2 = \pi \times 2r^2 h = 2 \times (\pi r^2 h) = 2V \). So the volume is doubled. So the answer is C.

Answer:

C) The volume is doubled.