Use the relationship among the formulas for the volumes of cones, cylinders, and spheres to solve the following problem: The volume of a cylinder is 24x−36 cm3 . What is the volume of a sphere if its radius is the same as the cylinder’s and the height of the cylinder is equal to the sphere’s diameter?
The formula for the volume of a cylinder is Vc = πr^2h, where r is the radius and h is the height of the cylinder.
We are given that the volume of the cylinder is 24x - 36 cm^3. Therefore, 24x - 36 = πr^2h.
We are also given that the height of the cylinder is equal to the sphere's diameter, so h = 2r.
Substituting h = 2r into the equation for the volume of the cylinder, we get:
24x - 36 = πr^2(2r)
24x - 36 = 2πr^3
Now, let's solve for the radius of the cylinder:
2πr^3 = 24x - 36
r^3 = (24x - 36)/(2π)
r^3 = 12x - 18)/π
Taking the cube root of both sides, we get:
r = (12x - 18)^(1/3)/π^(1/3)
The volume of a sphere is given by the formula Vs = (4/3)πr^3, where r is the radius of the sphere.
Substituting the value of r we found earlier into the formula for the volume of the sphere, we get:
Vs = (4/3)π((12x - 18)^(1/3)/π^(1/3))^3
Vs = (4/3)π(12x - 18)/(π)
Vs = (4/3)(12x - 18)
Vs = 16x - 24 cm^3
Therefore, the volume of the sphere is 16x - 24 cm^3.