Question
Look at this cone: 8 cm 9 cm If the radius is halved and the height is tripled, then which of the following statements about its volume will be true? also it has to be a fraction or a mixed fraction
Answers
GPT-4o mini
To analyze the change in volume when the radius and height of a cone are modified, we can start with the formula for the volume of a cone:
\[
V = \frac{1}{3} \pi r^2 h
\]
Given:
- The original radius \( r = 8 \) cm
- The original height \( h = 9 \) cm
1. **Calculate the original volume \( V \):**
\[
V = \frac{1}{3} \pi (8)^2 (9) = \frac{1}{3} \pi (64) (9) = \frac{1}{3} \pi (576) = 192\pi \, \text{cm}^3
\]
2. **Modify the dimensions:**
- Halve the radius: New radius \( r' = \frac{8}{2} = 4 \) cm
- Triple the height: New height \( h' = 3 \times 9 = 27 \) cm
3. **Calculate the new volume \( V' \):**
\[
V' = \frac{1}{3} \pi (4)^2 (27) = \frac{1}{3} \pi (16) (27) = \frac{1}{3} \pi (432) = 144\pi \, \text{cm}^3
\]
4. **Determine the ratio of the new volume to the original volume:**
\[
\frac{V'}{V} = \frac{144\pi}{192\pi} = \frac{144}{192} = \frac{3}{4}
\]
This shows that the new volume is \(\frac{3}{4}\) of the original volume.
Therefore, if the radius is halved and the height is tripled, the new volume is \(\frac{3}{4}\) of the original volume. Thus, the true statement about its volume is:
\[
\text{The new volume is } \frac{3}{4} \text{ of the original volume.}
\]
\[
V = \frac{1}{3} \pi r^2 h
\]
Given:
- The original radius \( r = 8 \) cm
- The original height \( h = 9 \) cm
1. **Calculate the original volume \( V \):**
\[
V = \frac{1}{3} \pi (8)^2 (9) = \frac{1}{3} \pi (64) (9) = \frac{1}{3} \pi (576) = 192\pi \, \text{cm}^3
\]
2. **Modify the dimensions:**
- Halve the radius: New radius \( r' = \frac{8}{2} = 4 \) cm
- Triple the height: New height \( h' = 3 \times 9 = 27 \) cm
3. **Calculate the new volume \( V' \):**
\[
V' = \frac{1}{3} \pi (4)^2 (27) = \frac{1}{3} \pi (16) (27) = \frac{1}{3} \pi (432) = 144\pi \, \text{cm}^3
\]
4. **Determine the ratio of the new volume to the original volume:**
\[
\frac{V'}{V} = \frac{144\pi}{192\pi} = \frac{144}{192} = \frac{3}{4}
\]
This shows that the new volume is \(\frac{3}{4}\) of the original volume.
Therefore, if the radius is halved and the height is tripled, the new volume is \(\frac{3}{4}\) of the original volume. Thus, the true statement about its volume is:
\[
\text{The new volume is } \frac{3}{4} \text{ of the original volume.}
\]