Question

Suppose a sphere, cylinder, and cone all share the same radius. The cylinder and cone also share the same height, which is twice the radius. Which of the following is true about the relationship between the volume formulas for the sphere, cylinder, and cone?(1 point)
Responses

cone = sphere − cylinder
cone = sphere − cylinder

cylinder = sphere − cone
cylinder = sphere − cone

sphere = cylinder + cone
sphere = cylinder + cone

cone = cylinder − sphere

Answer 1

To determine the correct relationship between the volumes of the sphere, cylinder, and cone, we first need to recall the formulas for the volumes of each shape:

1. **Sphere**:
\[
V = \frac{4}{3} \pi r^3
\]
2. **Cylinder**:
\[
V = \pi r^2 h
\]
(where \(h\) is the height)
3. **Cone**:
\[
V = \frac{1}{3} \pi r^2 h
\]

Given that the radius \( r \) is the same for all three shapes, and that the height \( h \) of both the cylinder and the cone is twice the radius (i.e., \( h = 2r \)), we can substitute this value into the volume formulas for the cylinder and the cone.

**Cylinder Volume**:
\[
V_{cylinder} = \pi r^2 (2r) = 2\pi r^3
\]

**Cone Volume**:
\[
V_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]

Now we can calculate the volume of the sphere, cylinder, and cone using these substitutions:

1. **Sphere Volume**:
\( V_{sphere} = \frac{4}{3} \pi r^3 \)

2. **Cylinder Volume**:
\( V_{cylinder} = 2\pi r^3 \)

3. **Cone Volume**:
\( V_{cone} = \frac{2}{3} \pi r^3 \)

Next, we check the provided options by expressing their relationships:

- Adding the volumes of the cylinder and cone:
\[
V_{cylinder} + V_{cone} = 2\pi r^3 + \frac{2}{3}\pi r^3 = \frac{6}{3}\pi r^3 + \frac{2}{3}\pi r^3 = \frac{8}{3}\pi r^3
\]
- Comparing this with the volume of the sphere:
\[
V_{sphere} = \frac{4}{3}\pi r^3
\]

We see:
- **Relation Check**:
\[
V_{cylinder} + V_{cone} = \frac{8}{3} \pi r^3 \text{ and } V_{sphere} = \frac{4}{3} \pi r^3
\]

None of the proposed responses based on the standard calculations seem valid. The actual relationship is:

- \( V_{cylinder} \) is greater than \( V_{sphere} \) up to some critical condition about proportions besides what’s shown in elementary geometry.

Among provided options, the closest true relation based on consistent formulations or aggregated results would be determined to narrow down in contexts which further boundaries and implications need to articulate distinctively cylinder and cone relationships relative to the original sphere.

However, no explicit true options were derived within your submitted quizzical bounds relating perfectly to each geometry foundational relationship you’ve structured around the identity deductively.

Conclusively, the formula that fits observed principles, disregarding numerical errors, aligns not triumphantly to any one concrete response stated. Be attentive to adjustments prompted against valid context geometrically applied. If we refine geometrical projection or aggregate characteristics, we can either reaffirm collective sums or differences eventually are structurally.

Since none of the expressions directly reflected calculations resulting straightforward; theoretically based on significant checkfunctions, a rational pause may yield specific results more conclusively checked with numerical approaches.