Question

The volume of a sphere is 72 m°. What is the volume of a cone if its radius is the same as the sphere's and its height is equal to the sphere's diameter?

Answer 1

[AI-generated answer — Bot GPT 3.5]

The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius of the sphere. In this case, the volume of the sphere is given as 72 m³.

72 = (4/3)πr^3

Multiply both sides by 3/4 to solve for πr^3:

πr^3 = (3/4) * 72
πr^3 = 54
r^3 = 54/π

Taking the cube root of both sides gives:

r = (54/π)^(1/3)

The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height. In this case, the radius of the cone (r) is the same as the radius of the sphere and the height (h) is equal to the sphere's diameter (2r).

Substituting the values:

V = (1/3)π * (54/π)^(1/3)^2 * 2(54/π)^(1/3)

Simplifying further:

V = (1/3)π * 54^(2/3) * 2 * 54^(1/3) * π^(2/3)

V = 2 * 18 * π * (54)^(1/3)

V ≈ 2 * 18 * π * 3.7797

V ≈ 406.18 m³

Therefore, the volume of the cone is approximately 406.18 m³.