The volume of a cone is ◂...▸ 25/3 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) Responses ◂...▸ 50/3 π cm3 Start Fraction 50 over 3 End Fraction pi cm cubed ◂...▸25 π cm3 25 pi cm cubed ◂...▸ 25/2 π cm3 Start Fraction 25 over 2 End Fraction pi cm cubed ◂...▸ 25/6 π cm3
The volume of a cone is given by V = (1/3)πr^2h, where r is the radius and h is the height.
Given that the volume of the cone is 25/3 π cm^3 and the radius is the same as the cone's height, we can say that r = h. Therefore, the volume of the cone can be written as V = (1/3)πh^3.
Since the height of the cone is equal to the diameter of the sphere, the radius of the sphere is equal to h/2.
The volume of a sphere is given by V = (4/3)πr^3, where r is the radius.
Substitute r = h/2 into the formula for the volume of the sphere:
V = (4/3)π(h/2)^3
V = (4/3)π(h^3/8)
V = (1/3)πh^3
V = 25/3 π cm^3
Therefore, the volume of the sphere is 25/3 π cm^3.
25/3 isnt a option
Apologies for the oversight. Let's calculate the correct volume of the sphere.
Given:
Volume of cone = 25/3 π cm^3
Volume of cone = (1/3)πr^2h
Volume of cone = (1/3)πr^2(2r) [Since h = 2r from the information given]
Solving for r in the cone volume equation:
25/3 π = (1/3)πr^2(2r)
25 = 2r^3
r^3 = 25/2
r = (25/2)^(1/3) = (25)^(1/3) / (2)^(1/3) = 5 / 2^(1/3) = 5∛(4)
Therefore, the radius of the cone is r = 5 / 2^(1/3).
The volume of a sphere with the same radius r is given by:
Volume of sphere = (4/3)πr^3
Volume of sphere = (4/3)π(5∛(4))^3
Volume of sphere = (4/3)π(125*4√4)
Volume of sphere = (4/3)π * 500
Volume of sphere = (2000/3)π
The volume of the sphere is 2000/3 π cm^3.