Question

The image shows three geometric shapes – a cone, a cylinder, and a sphere – each with a shaded circular cross-section of radius *r*.

The image contains three diagrams labeled A, B, and C. Diagram A depicts a cone with a circular base. The radius (*r*) of the base is indicated by a line segment. A dashed line shows the height of the cone, labeled as *3r*. The area of the circular base is shaded light blue. Diagram B shows a cylinder with a circular base and top. The radius (*r*) of both the base and top is indicated. The height of the cylinder is labeled as *3r*. The area of the circular base is shaded light blue. Diagram C shows a sphere with a diameter indicated by a dashed circle. Inside the sphere, a circular cross-section with radius *r* is shaded in light blue. The overall mood is purely informational and geometric; there is no emotional content. The diagrams are clean and simple, suggesting a mathematical or geometrical context, likely focused on volume or area calculations.

The volume of the cone is 25π.

Match the Question with the correct answer.

(5 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
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
How many times larger is the volume of the cylinder than the volume of the cone IF 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.

4r, 3r, 75π, 1/3, 75/2π, 50/3 π, 25π, 2r, 2, 100/3π, 3, 125π

Answer 1

To solve the questions given the information about the cone, cylinder, and sphere, we can use the formulas for their volumes:

1. **Volume of the cone \( V_c \)**:
\[
V_c = \frac{1}{3} \pi r^2 h
\]
Here, the height \( h \) of the cone is \( 3r \), so:
\[
V_c = \frac{1}{3} \pi r^2 (3r) = \pi r^3
\]
Given that \( V_c = 25\pi \), we can find \( r^3 \):
\[
\pi r^3 = 25\pi \implies r^3 = 25 \implies r = \sqrt[3]{25}
\]

2. **Volume of the sphere \( V_s \)**:
\[
V_s = \frac{4}{3} \pi r^3
\]
Substituting \( r^3 = 25 \):
\[
V_s = \frac{4}{3} \pi (25) = \frac{100}{3} \pi
\]

3. **Volume of the cylinder \( V_{cy}**:
\[
V_{cy} = \pi r^2 h
\]
The height of the cylinder is also \( 3r \):
\[
V_{cy} = \pi r^2 (3r) = 3\pi r^3
\]
Substituting \( r^3 = 25 \):
\[
V_{cy} = 3\pi (25) = 75\pi
\]

4. **Comparing the volumes**:
- **How many times larger is the volume of the sphere than the volume of the cone?**:
\[
\text{Ratio} = \frac{V_s}{V_c} = \frac{\frac{100}{3} \pi}{25 \pi} = \frac{100/3}{25} = \frac{100}{75} = \frac{4}{3} \approx 1.33
\]
So, the sphere is \(\frac{4}{3}\) times larger than the cone.

- **How many times larger is the volume of the cylinder than the volume of the cone?**:
\[
\text{Ratio} = \frac{V_{cy}}{V_c} = \frac{75\pi}{25\pi} = 3
\]
So, the cylinder is \(3\) times larger than the cone.

5. **In order for the volume of the cone + the volume of the sphere = the volume of the cylinder, the height would have to be**:
\[
V_c + V_s = V_{cy} \implies 25\pi + \frac{100}{3}\pi = 75\pi
\]
So, solving for height would suggest that the height must be adjusted.

Now, matching the responses:

1. The volume of the sphere if the radius is the same value as the cone: **100/3π**
2. 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? **4/3**
3. The volume of the cylinder: **75π**
4. How many times larger is the volume of the cylinder than the volume of the cone IF the cone and the cylinder had the same dimensions? **3**
5. In order for the volume of the cone + the volume of the sphere = the volume of the cylinder, the height would have to be **4r** (since we need the same dimensional relationships for the total volume).

Thus:

- Volume of the sphere: **100/3π**
- Sphere volume to cone volume ratio: **4/3**
- Volume of the cylinder: **75π**
- Cylinder volume to cone volume ratio: **3**
- Required height for equality: **4r**