Question
Cone: r 3h; Cylinder: r 3h; Sphere: r. The volume of the cone is 25 pi. Match the question with the correct answer: (How many times larger is the volume of the cylinder than the volume of the cone and the cylinder had the same dimensions?______; In order for the volume of the cone + the volume of the sphere = the volume of the cylinder, the height would have to be _____, not 3r for the cone and cylinder; The volume of the sphere if the radius is the same value as the cone _____; How many times larger is the volume of the sphere than the volume of the cone If the cone and the sphere had the same dimensions?_____; The volume of the cylinder _______)
Answers
Answered by
GPT-4o mini
To answer your questions, we'll need to calculate the volumes of the cone, cylinder, and sphere based on given dimensions.
1. **Volume of the Cone**: The volume of a cone is given by the formula:
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]
Given that the volume of the cone is \(25 \pi\), we have:
\[
\frac{1}{3} \pi r^2 h = 25 \pi
\]
Simplifying, we find:
\[
\frac{1}{3} r^2 h = 25 \implies r^2 h = 75
\]
2. **Volume of the Cylinder**: The volume of a cylinder is given by:
\[
V_{cylinder} = \pi r^2 h
\]
Given that the cone and cylinder have the same dimensions, we have:
\[
V_{cylinder} = \pi r^2 h = 3 \times (V_{cone}) = 3 \times 25\pi = 75\pi
\]
3. **Volume of the Sphere**: The volume of a sphere is given by:
\[
V_{sphere} = \frac{4}{3} \pi r^3
\]
We need to relate \(r^2 h = 75\) to the radius. To express \(h\) in terms of \(r\) for the cone, we can let \(h = \frac{75}{r^2}\).
4. **Finding the volume of the sphere**:
To get a numerical answer, we need to express \(r\) with respect to \(h\). However, let's calculate the volume now:
Since there is no specific value for \(r\) given, let's assume a general approach. If \(h = 3r\) (assuming the height of the cone and cylinder is three times the radius), we need to solve for \(r\):
Plug \(h\) into the equation from the cone:
\[
r^2 (3r) = 75 \implies 3r^3 = 75 \implies r^3 = 25 \implies r = \sqrt[3]{25}
\]
Now, we can calculate the volume of the sphere:
\[
V_{sphere} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (25)
\]
5. **Finally Answering the Questions**:
- **(How many times larger is the volume of the cylinder than the volume of the cone)**:
- Volume of cylinder / Volume of cone = \(75\pi / 25\pi = 3\).
- **(In order for the volume of the cone + the volume of the sphere = the volume of the cylinder, the height would have to be _____)**:
- You need to determine a new height based on the specific dimensions desired, but here you simply can write any value such that the resultant volume of sphere plus the cone is equal to \(75 \pi \), specifying that height should be based on specific dimensional ratios.
- **(The volume of the sphere if the radius is the same value as the cone)**:
- The volume will be \(\frac{4}{3} \pi (25) = \frac{100}{3} \pi\).
- **(How many times larger is the volume of the sphere than the volume of the cone If the cone and the sphere had the same dimensions)**:
- Volume of sphere / Volume of cone = \((\frac{100}{3}\pi) / (25\pi) = \frac{4}{3}\) times larger.
- **(The volume of the cylinder)**:
- The volume of the cylinder is \(75\pi\).
Based on the analysis above, here are the answers matched to the questions:
1. \(3\)
2. Height should be \(h\) if we solve for a specific condition of equality.
3. \(\frac{100}{3} \pi\)
4. \(\frac{4}{3}\)
5. \(75\pi\)
1. **Volume of the Cone**: The volume of a cone is given by the formula:
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]
Given that the volume of the cone is \(25 \pi\), we have:
\[
\frac{1}{3} \pi r^2 h = 25 \pi
\]
Simplifying, we find:
\[
\frac{1}{3} r^2 h = 25 \implies r^2 h = 75
\]
2. **Volume of the Cylinder**: The volume of a cylinder is given by:
\[
V_{cylinder} = \pi r^2 h
\]
Given that the cone and cylinder have the same dimensions, we have:
\[
V_{cylinder} = \pi r^2 h = 3 \times (V_{cone}) = 3 \times 25\pi = 75\pi
\]
3. **Volume of the Sphere**: The volume of a sphere is given by:
\[
V_{sphere} = \frac{4}{3} \pi r^3
\]
We need to relate \(r^2 h = 75\) to the radius. To express \(h\) in terms of \(r\) for the cone, we can let \(h = \frac{75}{r^2}\).
4. **Finding the volume of the sphere**:
To get a numerical answer, we need to express \(r\) with respect to \(h\). However, let's calculate the volume now:
Since there is no specific value for \(r\) given, let's assume a general approach. If \(h = 3r\) (assuming the height of the cone and cylinder is three times the radius), we need to solve for \(r\):
Plug \(h\) into the equation from the cone:
\[
r^2 (3r) = 75 \implies 3r^3 = 75 \implies r^3 = 25 \implies r = \sqrt[3]{25}
\]
Now, we can calculate the volume of the sphere:
\[
V_{sphere} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (25)
\]
5. **Finally Answering the Questions**:
- **(How many times larger is the volume of the cylinder than the volume of the cone)**:
- Volume of cylinder / Volume of cone = \(75\pi / 25\pi = 3\).
- **(In order for the volume of the cone + the volume of the sphere = the volume of the cylinder, the height would have to be _____)**:
- You need to determine a new height based on the specific dimensions desired, but here you simply can write any value such that the resultant volume of sphere plus the cone is equal to \(75 \pi \), specifying that height should be based on specific dimensional ratios.
- **(The volume of the sphere if the radius is the same value as the cone)**:
- The volume will be \(\frac{4}{3} \pi (25) = \frac{100}{3} \pi\).
- **(How many times larger is the volume of the sphere than the volume of the cone If the cone and the sphere had the same dimensions)**:
- Volume of sphere / Volume of cone = \((\frac{100}{3}\pi) / (25\pi) = \frac{4}{3}\) times larger.
- **(The volume of the cylinder)**:
- The volume of the cylinder is \(75\pi\).
Based on the analysis above, here are the answers matched to the questions:
1. \(3\)
2. Height should be \(h\) if we solve for a specific condition of equality.
3. \(\frac{100}{3} \pi\)
4. \(\frac{4}{3}\)
5. \(75\pi\)