Question

The relationships of the volumes of cylinders, cones, and spheres (considering that the height and radius are the same for all of the objects):

Put the items in order from greatest amount of cones needed to least amount of cones needed.



Greatest amount of cones needed



Least amount of cones needed

Options (OUT OF ORDER): Number of Cones needed, along with 1 sphere to equal the volume of one cylinder. Number of Cones needed to equal the volume of one Sphere. Number of Cones needed to equal the volume of one Cylinder.

Answer 1

To compare the volumes of cylinders, cones, and spheres with the same height (h) and radius (r), we can use the formulas for the volumes of each shape:

1. **Volume of a Cylinder (V_c)**:
\( V_c = \pi r^2 h \)

2. **Volume of a Cone (V_co)**:
\( V_{co} = \frac{1}{3} \pi r^2 h \)

3. **Volume of a Sphere (V_s)**:
\( V_s = \frac{4}{3} \pi r^3 \)

### Calculating the number of cones needed:

To find how many cones are needed to equal the volume of one cylinder:

\[
\text{Number of cones needed to equal the volume of one cylinder} = \frac{V_c}{V_{co}} = \frac{\pi r^2 h}{\frac{1}{3} \pi r^2 h} = 3
\]

To find how many cones are needed to equal the volume of one sphere, we need to compare the volume of the sphere and the cone:

### Step 1: Calculate how many cones we need to equal the volume of one sphere:

\[
\text{Number of cones needed to equal the volume of one sphere} = \frac{V_s}{V_{co}} = \frac{\frac{4}{3} \pi r^3}{\frac{1}{3} \pi r^2 h} = \frac{4r}{h}
\]

### Conclusion on ordering:
- The number of cones needed to equal the volume of one cylinder is **3**.
- The number of cones needed to equal the volume of one sphere is dependent on the ratio \(\frac{4r}{h}\).

#### Now for ordering:
1. If \(h\) is less than \(4r\), \( \frac{4r}{h} > 3 \) (more cones needed than for one cylinder).
2. If \(h = 4r\), \( \frac{4r}{h} = 3 \) (same amount of cones needed).
3. If \(h\) is greater than \(4r\), \( \frac{4r}{h} < 3 \) (fewer cones needed than for one cylinder).

Given that we don't have a specific value for \(h\), we categorize based on general assumptions:

- **Greatest amount of cones needed**: This occurs when \(h < 4r\) (more than 3 cones needed compared to the sphere).
- **Middle**: When \(h = 4r\) (exactly 3 cones needed which equals the cylinder).
- **Least amount of cones needed**: Occurs when \(h > 4r\) (fewer than 3 cones needed).

### Final ordering without specific values:
1. **Number of cones needed to equal the volume of one sphere** (potentially most depending on h/r ratio).
2. **Number of cones needed to equal the volume of one cylinder** (3 cones).
3. **Number of cones needed to equal the volume of one sphere** (fewer than 3 when h > 4r).

Put in order from greatest amount of cones needed to least amount:
1. Number of cones needed to equal the volume of one sphere (potentially the most).
2. Number of cones needed to equal the volume of one cylinder (3 cones).
3. Number of cones needed to equal the volume of one cone (the least).

In summary, the exact ordering can depend on the specific values of height and radius, mainly determined by the ratio \( \frac{h}{r} \).