The volume of a cone is 25 π cm3 . What is the volume of a sphere if its radius is the same as the cone’s and the height of the cone is equal to the sphere’s diameter?(1 point)
25 π cm3
25/2 π cm3
50/3 π cm3
25/6 π cm3
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height. Since the volume of the cone is 25π cm^3, we know that V = 25π.
The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius. In this case, the radius of the sphere is the same as the cone's, so r = r. The height of the cone is equal to the sphere's diameter, so h = 2r.
Substitute h = 2r into the formula for the volume of the cone:
V = (1/3)πr^2(2r)
V = (2/3)πr^3.
Since the volume of the cone is given as 25π, we have:
25π = (2/3)πr^3
r^3 = (25/3)(π/π)
r^3 = 25/3
r = (25/3)^(1/3) = 2.924 cm.
Now, substitute r = 2.924 into the formula for the volume of the sphere:
V = (4/3)π(2.924)^3
V = (4/3)π(24.37096)
V = 109.953 cm^3.
Therefore, the volume of the sphere is approximately 109.953 cm^3.