To find the volume of a sphere, we need to use the formula V = (4/3)πr³, where r is the radius of the sphere. In this case, the radius of the sphere is the same as the radius of the cone, and the height of the cone is equal to the diameter of the sphere.
Given that the volume of the cone is 25/3 π^3, we can find the radius using the formula V = (1/3)πr²h, where V is the volume, r is the radius, and h is the height.
Using the volume of the cone, we have (1/3)πr²h = 25/3 π^3.
Simplifying, we have r²h = 25 π^2.
Since the height of the cone is equal to the diameter of the sphere, we can write h = 2r.
Substituting into the equation, we have r²(2r) = 25 π^2.
This simplifies to 2r³ = 25 π^2.
Dividing both sides by 2, we get r³ = 25 π^2 / 2.
Finally, taking the cube root of both sides to solve for r, we have r = (25 π^2 / 2)^(1/3).
Now that we know the radius of the sphere, we can find its volume using the formula V = (4/3)πr³.
Substituting the value of r into the equation, we have V = (4/3)π[(25 π^2 / 2)^(1/3)]³.
Simplifying this expression gives V = (4/3)π(25 π^2 / 2)^(3/3).
Simplifying further, we have V = (25/6)Ï€^3.
Therefore, the volume of the sphere is 25/6 π^3.