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)%0D%0AResponses%0D%0A%0D%0A256 π cm3%0D%0AStart Fraction 25 over 6 End Fraction pi cm cubed%0D%0A%0D%0A25 π cm3%0D%0A25 pi cm cubed%0D%0A%0D%0A503 π cm3%0D%0AStart Fraction 50 over 3 End Fraction pi cm cubed%0D%0A%0D%0A252 π cm3
The volume of a cone is given by the formula V = (1/3)πr²h, where r is the radius and h is the height.
Given that the volume of the cone is 25/3 π cm³, we can set up the equation:
(1/3)πr²h = 25/3 π
Since the height of the cone is equal to the sphere's diameter, we can rewrite the equation as:
(1/3)πr²(2r) = 25/3 π
Simplifying the equation, we have:
(2/3)πr³ = 25/3 π
Multiplying both sides by 3/π, we get:
2r³ = 25
Taking the cube root of both sides, we find:
r = ∛(25/2)
Now, the volume of a sphere is given by the formula V = (4/3)πr³, where r is the radius.
Substituting the value of r we found, we have:
V = (4/3)π(∛25/2)³
Simplifying, we get:
V ≈ 256π/3 cm³
Therefore, the volume of the sphere is approximately 256π cm³.