The volume of a cone is 253 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
25 π cm3
25 pi cm cubed
252 π cm3
Start Fraction 25 over 2 End Fraction pi cm cubed
256 π cm3
Start Fraction 25 over 6 End Fraction pi cm cubed
503 π cm3
Start Fraction 50 over 3 End Fraction pi cm cubed
The volume of a cone is given by the formula V_cone = (1/3) * π * r^2 * h, where r is the radius and h is the height of the cone.
We are given that V_cone = 253π cm^3. Since the radius of the sphere is the same as the cone's and the height of the cone is equal to the sphere's diameter, we can say that the radius and the diameter of the sphere are both equal to r.
The volume of a sphere is given by the formula V_sphere = (4/3) * π * r^3. Substituting r=r into the formula, we get V_sphere = (4/3) * π * r^3 = (4/3) * π * r * r * r = 4π * r^3/3.
Since the radius of the sphere is the same as the cone's, we can equate their volumes:
253π = 4π * r^3/3.
Dividing both sides of the equation by 4π, we get r^3/3 = 253/4. Multiplying both sides of the equation by 3, we get r^3 = (253/4) * 3.
Simplifying the right side of the equation, we get r^3 = 253/4 * 3 = 759/4.
Taking the cube root of both sides, we get r = ∛(759/4).
The volume of the sphere is given by V_sphere = (4/3) * π * r^3 = (4/3) * π * (∛(759/4))^3 = (4/3) * π * (759/4) = (3/4) * 759π = 569.25π.
So, the volume of the sphere is 569.25π cm^3, which is approximately equal to 1788.17 cm^3.
Therefore, none of the given options is correct.