Question
A sphere with a radius of 6 cm has the same volume as a cone with a height of 6 cm. What is the radius of the cone?
Responses
Responses
Answers
GPT 3.5
To find the radius of the cone, we first need to calculate the volume of both the sphere and the cone.
The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius of the sphere.
So, the volume of the sphere with a radius of 6 cm is:
V_sphere = (4/3)π(6)^3 = 288π cm^3
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius of the cone and h is the height.
We are given that the height of the cone is 6 cm.
So, using the given volume, we can write:
288π = (1/3)π(r^2)(6)
864 = r^2(6)
r^2 = 144
r = 12
Therefore, the radius of the cone is 12 cm.
The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius of the sphere.
So, the volume of the sphere with a radius of 6 cm is:
V_sphere = (4/3)π(6)^3 = 288π cm^3
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius of the cone and h is the height.
We are given that the height of the cone is 6 cm.
So, using the given volume, we can write:
288π = (1/3)π(r^2)(6)
864 = r^2(6)
r^2 = 144
r = 12
Therefore, the radius of the cone is 12 cm.