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?(1 point)

Responses

2 spheres = 1 cone
2 spheres = 1 cone

2 cylinders = 1 cone
2 cylinders = 1 cone

2 cones = 1 sphere
2 cones = 1 sphere

2 cones = 1 cylinder

1 answer

To analyze the relationship between the volumes of a sphere, cylinder, and cone with the same radius (r), and the cylinder and cone having a height (h) that is twice the radius, we start by calculating the volumes using their respective formulas.

  1. Volume of the Sphere (V_s): \[ V_s = \frac{4}{3} \pi r^3 \]

  2. Volume of the Cylinder (V_c): The height of the cylinder is \(h = 2r\). \[ V_c = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3 \]

  3. Volume of the Cone (V_n): The height of the cone is also \(h = 2r\). \[ V_n = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]

Now we can compare the volumes:

  • Relationship between 2 spheres and the cone: \[ 2V_s = 2 \left(\frac{4}{3} \pi r^3\right) = \frac{8}{3} \pi r^3 \quad \text{and} \quad V_n = \frac{2}{3} \pi r^3 \] So, \(2V_s \neq V_n\).

  • Relationship between 2 cylinders and the cone: \[ 2V_c = 2(2\pi r^3) = 4\pi r^3 \quad \text{and} \quad V_n = \frac{2}{3} \pi r^3 \] So, \(2V_c \neq V_n\).

  • Relationship between 2 cones and the sphere: \[ 2V_n = 2\left(\frac{2}{3} \pi r^3\right) = \frac{4}{3} \pi r^3 \quad \text{and} \quad V_s = \frac{4}{3} \pi r^3 \] So, \(2V_n = V_s\).

  • Relationship between 2 cones and the cylinder: \[ 2V_n = 2 \left(\frac{2}{3} \pi r^3\right) = \frac{4}{3} \pi r^3 \quad \text{and} \quad V_c = 2\pi r^3 \] So, \(2V_n \neq V_c\).

Now we can conclude:

The only true statement is: 2 cones = 1 sphere.