suppose a sphere, cylinder and cone all share the same radius and 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?

Answer 1

To analyze the relationship between the volumes of the sphere, cylinder, and cone given that they share the same radius \( r \), and the height of both the cylinder and the cone is twice the radius (\( h = 2r \)), we can use the volume formulas for each shape.

1. Volume of a Sphere: \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]

2. Volume of a Cylinder: The volume of a cylinder is given by: \[ V_{\text{cylinder}} = \pi r^2 h \] Substituting \( h = 2r \): \[ V_{\text{cylinder}} = \pi r^2 (2r) = 2\pi r^3 \]

3. Volume of a Cone: The volume of a cone is given by: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] Again, substituting \( h = 2r \): \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]

Now we have the volumes:

• \( V_{\text{sphere}} = \frac{4}{3} \pi r^3 \)
• \( V_{\text{cylinder}} = 2\pi r^3 \)
• \( V_{\text{cone}} = \frac{2}{3} \pi r^3 \)

Let's compare these volumes:

• The volume of the cylinder \( V_{\text{cylinder}} \) is: \[ 2\pi r^3 = \frac{6}{3}\pi r^3 \] Therefore, \[ V_{\text{cylinder}} > V_{\text{sphere}} \text{ (since \( \frac{6}{3} > \frac{4}{3} \))} \]

• The volume of the cone \( V_{\text{cone}} \) is: \[ \frac{2}{3}\pi r^3 \] Therefore, \[ V_{\text{cone}} < V_{\text{sphere}} \text{ (since \( \frac{2}{3} < \frac{4}{3} \))} \]

In conclusion, we have:

• \( V_{\text{sphere}} > V_{\text{cone}} \)
• \( V_{\text{cylinder}} > V_{\text{sphere}} \)
• \( V_{\text{cylinder}} > V_{\text{cone}} \)

Thus, the relationship among the volumes is: \[ V_{\text{cone}} < V_{\text{sphere}} < V_{\text{cylinder}} \]

Answer 2

Which answer is it?

A. sphere - cone = cylinder.
B. sphere = cylinder + cone.
C. 3 cones = 1 cylinder.
D. 1 cylinder = 2 cones + 1 sphere

Answer 3

To determine which relationship is true based on the volumes we found earlier, let's summarize the volumes:

• Volume of the sphere:
  \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]

• Volume of the cylinder:
  \[ V_{\text{cylinder}} = 2\pi r^3 \]

• Volume of the cone:
  \[ V_{\text{cone}} = \frac{2}{3} \pi r^3 \]

Now let's check each of the options based on these volume formulas:

A. \( \text{sphere} - \text{cone} = \text{cylinder} \)

• \( \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{2}{3} \pi r^3 \)
• This is false because \( \frac{2}{3} \pi r^3 \) is \( V_{\text{cone}} \), not \( V_{\text{cylinder}} \).

B. \( \text{sphere} = \text{cylinder} + \text{cone} \)

• \( \frac{4}{3} \pi r^3 = 2\pi r^3 + \frac{2}{3} \pi r^3 \)
• On the right side:
  • \( 2\pi r^3 = \frac{6}{3} \pi r^3\)
  • Adding \( \frac{2}{3} \pi r^3 \) gives \( \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{8}{3} \pi r^3 \)
• This is false since \( \frac{4}{3} \pi r^3 \neq \frac{8}{3} \pi r^3 \).

C. \( 3 \text{ cones} = 1 \text{ cylinder} \)

• \( 3 \cdot \frac{2}{3} \pi r^3 = 2\pi r^3 \)
• \( 3 \cdot \frac{2}{3} = 2 \), so this is true because both sides equal \( 2\pi r^3 \).

D. \( 1 \text{ cylinder} = 2 \text{ cones} + 1 \text{ sphere} \)

• This expands to \( 2\pi r^3 = 2 \cdot \frac{2}{3} \pi r^3 + \frac{4}{3} \pi r^3 \)
• On the right:
  • \( 2 \cdot \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3 \)
  • Adding \( \frac{4}{3} \pi r^3 + \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \)
• This is false because \( 2\pi r^3 \neq \frac{8}{3} \pi r^3 \).

Conclusion:

The correct answer is C. 3 cones = 1 cylinder.