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)
Let's define the radius of all three objects as r, and since the height of both the cylinder and cone is given as twice the radius, we'll define the height (h) as 2r for both.
The volume formulas for these three-dimensional objects are as follows:
1. The volume of a sphere (V_sphere) is given by the formula:
\[ V_{sphere} = \frac{4}{3}\pi r^3 \]
2. The volume of a cylinder (V_cylinder) is given by the formula:
\[ V_{cylinder} = \pi r^2 h \]
Since the height h is equal to 2r, this becomes:
\[ V_{cylinder} = \pi r^2 (2r) = 2\pi r^3 \]
3. The volume of a cone (V_cone) is given by the formula:
\[ V_{cone} = \frac{1}{3}\pi r^2 h \]
Again, since the height h is equal to 2r, this becomes:
\[ V_{cone} = \frac{1}{3}\pi r^2 (2r) = \frac{2}{3}\pi r^3 \]
Now, let's examine their relationships:
- Comparing the volume of the sphere and the cylinder, we see that the volume of the sphere is \( \frac{4}{3} \) times the volume of the cylinder:
\[ V_{sphere} = \frac{4}{3}\pi r^3 = \frac{2}{3} V_{cylinder} \]
- Comparing the volume of the cylinder to the cone, the cylinder's volume is 3 times the cone's volume. This is because the cone's volume is a third of the cylinder volume with the same base and height:
\[ V_{cylinder} = 3 V_{cone} \]
- Finally, the volume of the sphere is double the volume of the cone:
\[ V_{sphere} = 2 V_{cone} \]
So altogether, we have the following relationships between the volumes:
\[ V_{cylinder} = 3 V_{cone} \]
\[ V_{sphere} = 2 V_{cone} = \frac{2}{3} V_{cylinder} \]
Therefore, the sphere will have a volume that is two-thirds that of the cylinder and double that of the cone, given the specified dimensions.