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
252 π cm3
Start Fraction 25 over 2 End Fraction pi cm cubed
25 π cm3
25 pi cm cubed
503 π cm3
Start Fraction 50 over 3 End Fraction pi cm cubed
256 π cm3
Let's first find the radius and height of the cone.
The volume of a cone is given by the formula:
V = (1/3) * π * r^2 * h,
where V is the volume, π is pi, r is the radius, and h is the height.
We are given that the volume of the cone is 25/3 π cm^3, so we can set up the equation:
25/3 π = (1/3) * π * r^2 * h.
Since the radius of the cone is the same as the radius of the sphere, let's call it r.
We are also given that the height of the cone is equal to the sphere's diameter, so let's call it d.
Since the diameter is twice the radius, we have d = 2r.
Now we can rewrite the equation:
25/3 π = (1/3) * π * r^2 * d.
Substituting d = 2r, we get:
25/3 π = (1/3) * π * r^2 * 2r.
Simplifying, we have:
25/3 π = (2/3) * π * r^3.
Cancelling out the pi, we have:
25/3 = (2/3) * r^3.
Now we can solve for r:
r^3 = (25/3) * (3/2) = 25/2.
Taking the cube root of both sides, we get:
r = (25/2)^(1/3) = (5^3/2^3)^(1/3) = 5/2.
Now we can find the volume of the sphere.
The volume of a sphere is given by the formula:
V = (4/3) * π * r^3.
Substituting r = 5/2, we get:
V = (4/3) * π * (5/2)^3 = (4/3) * π * (125/8) = (500/24) * π.
Thus, the volume of the sphere is 500/24 π cm^3.
The correct answer is not among the given options.