Suppose a sphere, cylinder, and a cone all share the same radius. The cylinder and the 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?
Let's define the formulas for the volume of each shape, given that the cylinder and cone have the same height (h) and that the height is twice the radius (r), so h = 2r. For all three shapes, we'll use r for the radius.
1. Sphere Volume (V_sphere):
The formula for the volume of a sphere is given by:
\[ V_{sphere} = \frac{4}{3} \pi r^3 \]
2. Cylinder Volume (V_cylinder):
The volume of a cylinder is given by:
\[ V_{cylinder} = \pi r^2 h \]
Since h = 2r, we substitute to get:
\[ V_{cylinder} = \pi r^2 (2r) = 2\pi r^3 \]
3. Cone Volume (V_cone):
The volume of a cone is one third of the volume of a cylinder with the same base and height:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]
Since h = 2r, we substitute to get:
\[ V_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now, let's compare these formulas:
- The sphere's volume is \( \frac{4}{3} \pi r^3 \).
- The cylinder's volume is \( 2\pi r^3 \), which is three times the cone's volume.
- The cone's volume is \( \frac{2}{3} \pi r^3 \), which is one-third of the cylinder's volume and half the sphere's volume.
So the volumes relate to each other like this:
- The cylinder's volume is 3 times the cone's volume.
- The sphere's volume is 2 times the cone's volume.
- The cylinder's volume is 1.5 times the sphere's volume.
Putting these relationships together:
- \( V_{cylinder} = 3 \times V_{cone} \)
- \( V_{sphere} = 2 \times V_{cone} \)
- \( V_{cylinder} = 1.5 \times V_{sphere} \)