Question

To answer this question, let's first write down the volume formulas for the sphere, cylinder, and cone based on the given dimensions.

1. Sphere: The volume \( V_s \) of a sphere with radius \( r \) is given by: \[ V_s = \frac{4}{3} \pi r^3 \]

2. Cylinder: The volume \( V_c \) of a cylinder with radius \( r \) and height \( h = 2r \) is given by: \[ V_c = \pi r^2 h = \pi r^2 (2r) = 2 \pi r^3 \]

3. Cone: The volume \( V_{cone} \) of a cone with radius \( r \) and height \( h = 2r \) is given by: \[ V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]

Now we can compare the volumes:

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

• Volume of the cylinder: \[ V_c = 2 \pi r^3 \]

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

Next, we can see how these formulas compare to each other:

1. To check if \( V_s = V_c + V_{cone} \): \[ V_c + V_{cone} = 2 \pi r^3 + \frac{2}{3} \pi r^3 = \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{8}{3} \pi r^3 \] This is not equal to \( V_s \).

2. To check if \( V_{cone} = V_c - V_s \): \[ V_c - V_s = 2 \pi r^3 - \frac{4}{3} \pi r^3 = \frac{6}{3} \pi r^3 - \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3 = V_{cone} \] This statement holds true.

3. To check if \( V_{cone} = V_s - V_c \): \[ V_s - V_c = \frac{4}{3} \pi r^3 - 2 \pi r^3 = \frac{4}{3} \pi r^3 - \frac{6}{3} \pi r^3 = -\frac{2}{3} \pi r^3 \] This does not hold.

4. To check if \( V_c = V_s - V_{cone} \): \[ V_s - V_{cone} = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{2}{3} \pi r^3 \] This does not hold.

Based on the calculations, the relationship that holds true is:

cone = cylinder – sphere.

So, the correct response is: cone = cylinder – sphere.