the volume of a cone is 25/3 (pi symbol) cm^3. 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
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height of the cone.
We are given that the volume of the cone is 25/3 π cm^3. Therefore, we can write the equation:
25/3 π = (1/3)π(r^2)(h)
Since the height of the cone is equal to the diameter of the sphere, we can write h = 2r.
Substituting this into the equation, we have:
25/3 π = (1/3)π(r^2)(2r)
Simplifying the equation, we get:
25/3 = 2r^3
Multiplying both sides of the equation by 3 and dividing by 2, we get:
r^3 = (25/3)(3/2) = 25/2
Taking the cube root of both sides, we find:
r = ∛(25/2)
The volume of a sphere is given by the formula V = (4/3)πr^3.
Substituting the value of r we found into the equation, we have:
V = (4/3)π(∛(25/2))^3
Simplifying the equation, we get:
V = (4/3)π(5∛2)^3
V = (4/3)π(125√2)
V = (500/3)π√2
Therefore, the volume of the sphere is (500/3)π√2 cm^3.