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
To determine which of the given relationships is true, we need to examine the volume formulas for a sphere, cylinder, and cone.
The formulas for the volumes are as follows:
Sphere: \( V_{sphere} = \frac{4}{3}\pi r^3 \)
Cylinder: \( V_{cylinder} = \pi r^2 h \)
Cone: \( V_{cone} = \frac{1}{3}\pi r^2 h \)
Given that the cylinder and cone have the same height, which is twice the radius, we can write the height as \( h = 2r \).
Now, we substitute this height into the formulas for the cylinder and the cone:
Cylinder: \( V_{cylinder} = \pi r^2 (2r) = 2\pi r^3 \)
Cone: \( V_{cone} = \frac{1}{3}\pi r^2 (2r) = \frac{2}{3}\pi r^3 \)
Notice that the volume of the cylinder is exactly three times the volume of the cone (\( V_{cylinder} = 3 \cdot V_{cone} \)).
It's now time to inspect the given options and to compare them to the volume formulas. Considering the volumes we've found:
- The sphere's volume does not equal the cylinder's volume plus the cone's volume (eliminating the third option).
- The cone's volume is not equivalent to the cylinder's volume minus the sphere's volume (eliminating the fourth option).
To determine if the first or second options are true, let's calculate the difference between the volumes of the sphere and the cylinder:
\( V_{sphere} - V_{cylinder} = \frac{4}{3}\pi r^3 - 2\pi r^3 = \frac{4}{3}\pi r^3 - \frac{6}{3}\pi r^3 = -\frac{2}{3}\pi r^3 \)
This result is the negative of the volume of the cone, not the positive. Therefore, none of the provided responses are correct. If the equation were to be rearranged to \( V_{cylinder} + V_{cone} = V_{sphere} \), it would accurately represent the relationship between the three volumes. However, such an option is not presented.