The volume of a cone is 25/3 π cm^3 . 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?
First, we need to find the radius and height of the cone.
The formula for the volume of a cone is given by V = (1/3)πr^2h, where r is the radius of the base of the cone and h is the height of the cone.
Given that V = 25/3 π cm^3, we can replace V in the equation and solve for r and h:
25/3 = (1/3)πr^2h
25 = r^2h
Given that the height of the cone is equal to the sphere's diameter, and since the diameter of a sphere is twice its radius, h = 2r.
Substitute h = 2r into 25 = r^2h:
25 = r^2(2r)
25 = 2r^3
r^3 = 25/2
r = ∛(25/2) = 5/∛2
Now that we have found the radius of the cone, we can find the volume of the sphere.
The volume of a sphere is given by V = (4/3)πr^3.
Substitute r = 5/∛2 into V:
V = (4/3)π(5/∛2)^3
V = (4/3)π(125/8∛2)
V = 500/24π∛2
V ≈ 327.77 cm^3
Therefore, the volume of the sphere with a radius equal to the cone's and a height equal to the sphere's diameter is approximately 327.77 cm^3.