Question
Put the cones in order from greatest volume ( put at the top of the list) to least volume (put at the bottom of the list): (radius is 8 cm/height 18 cm; radius is 6 cm/height 9 cm; radius is 5 m/ height is 13 m; radius is 3 cm/ height is 21 cm; radius is 3 cm/ height is 8 cm)
Answers
GPT-4o mini
To compare the volumes of the cones, we can use the formula for the volume of a cone:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.
Let's calculate the volumes of each cone:
1. **Cone with radius 8 cm and height 18 cm**:
\[
V_1 = \frac{1}{3} \pi (8^2)(18) = \frac{1}{3} \pi (64)(18) = \frac{1}{3} \pi (1152) = 384\pi \, \text{cm}^3
\]
2. **Cone with radius 6 cm and height 9 cm**:
\[
V_2 = \frac{1}{3} \pi (6^2)(9) = \frac{1}{3} \pi (36)(9) = \frac{1}{3} \pi (324) = 108\pi \, \text{cm}^3
\]
3. **Cone with radius 5 m and height 13 m**:
\[
V_3 = \frac{1}{3} \pi (5^2)(13) = \frac{1}{3} \pi (25)(13) = \frac{1}{3} \pi (325) = \frac{325}{3}\pi \, \text{m}^3 \approx 108.33\pi \, \text{m}^3
\]
(Note that we need to convert cm³ to m³ for comparison with this cone. There are \( 1,000,000 \, \text{cm}^3 \) in \( 1 \, \text{m}^3 \), so we multiply \( 108.33 \pi \) by \( 10^6 \) to get volume in cm³: \( 108.33 \times 10^6 \pi \, \text{cm}^3 \)).
4. **Cone with radius 3 cm and height 21 cm**:
\[
V_4 = \frac{1}{3} \pi (3^2)(21) = \frac{1}{3} \pi (9)(21) = \frac{1}{3} \pi (189) = 63\pi \, \text{cm}^3
\]
5. **Cone with radius 3 cm and height 8 cm**:
\[
V_5 = \frac{1}{3} \pi (3^2)(8) = \frac{1}{3} \pi (9)(8) = \frac{1}{3} \pi (72) = 24\pi \, \text{cm}^3
\]
Now we can list the calculated volumes (in terms of \(\pi\) for comparison):
- \( V_1 = 384\pi \, \text{cm}^3 \)
- \( V_2 \approx 108\pi \, \text{cm}^3 \)
- \( V_3 \approx 108.33 \times 10^6 \pi \, \text{cm}^3 \)
- \( V_4 = 63\pi \, \text{cm}^3 \)
- \( V_5 = 24\pi \, \text{cm}^3 \)
Sorting from greatest volume to least:
1. Cone with radius 5 m and height 13 m \(\approx 108.33 \times 10^6 \pi\)
2. Cone with radius 8 cm and height 18 cm \(384\pi\)
3. Cone with radius 6 cm and height 9 cm \(108\pi\)
4. Cone with radius 3 cm and height 21 cm \(63\pi\)
5. Cone with radius 3 cm and height 8 cm \(24\pi\)
So the final order is:
1. Radius 5 m / Height 13 m
2. Radius 8 cm / Height 18 cm
3. Radius 6 cm / Height 9 cm
4. Radius 3 cm / Height 21 cm
5. Radius 3 cm / Height 8 cm
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.
Let's calculate the volumes of each cone:
1. **Cone with radius 8 cm and height 18 cm**:
\[
V_1 = \frac{1}{3} \pi (8^2)(18) = \frac{1}{3} \pi (64)(18) = \frac{1}{3} \pi (1152) = 384\pi \, \text{cm}^3
\]
2. **Cone with radius 6 cm and height 9 cm**:
\[
V_2 = \frac{1}{3} \pi (6^2)(9) = \frac{1}{3} \pi (36)(9) = \frac{1}{3} \pi (324) = 108\pi \, \text{cm}^3
\]
3. **Cone with radius 5 m and height 13 m**:
\[
V_3 = \frac{1}{3} \pi (5^2)(13) = \frac{1}{3} \pi (25)(13) = \frac{1}{3} \pi (325) = \frac{325}{3}\pi \, \text{m}^3 \approx 108.33\pi \, \text{m}^3
\]
(Note that we need to convert cm³ to m³ for comparison with this cone. There are \( 1,000,000 \, \text{cm}^3 \) in \( 1 \, \text{m}^3 \), so we multiply \( 108.33 \pi \) by \( 10^6 \) to get volume in cm³: \( 108.33 \times 10^6 \pi \, \text{cm}^3 \)).
4. **Cone with radius 3 cm and height 21 cm**:
\[
V_4 = \frac{1}{3} \pi (3^2)(21) = \frac{1}{3} \pi (9)(21) = \frac{1}{3} \pi (189) = 63\pi \, \text{cm}^3
\]
5. **Cone with radius 3 cm and height 8 cm**:
\[
V_5 = \frac{1}{3} \pi (3^2)(8) = \frac{1}{3} \pi (9)(8) = \frac{1}{3} \pi (72) = 24\pi \, \text{cm}^3
\]
Now we can list the calculated volumes (in terms of \(\pi\) for comparison):
- \( V_1 = 384\pi \, \text{cm}^3 \)
- \( V_2 \approx 108\pi \, \text{cm}^3 \)
- \( V_3 \approx 108.33 \times 10^6 \pi \, \text{cm}^3 \)
- \( V_4 = 63\pi \, \text{cm}^3 \)
- \( V_5 = 24\pi \, \text{cm}^3 \)
Sorting from greatest volume to least:
1. Cone with radius 5 m and height 13 m \(\approx 108.33 \times 10^6 \pi\)
2. Cone with radius 8 cm and height 18 cm \(384\pi\)
3. Cone with radius 6 cm and height 9 cm \(108\pi\)
4. Cone with radius 3 cm and height 21 cm \(63\pi\)
5. Cone with radius 3 cm and height 8 cm \(24\pi\)
So the final order is:
1. Radius 5 m / Height 13 m
2. Radius 8 cm / Height 18 cm
3. Radius 6 cm / Height 9 cm
4. Radius 3 cm / Height 21 cm
5. Radius 3 cm / Height 8 cm