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
To solve this problem, we need to know the relationship between the formulas for the volumes of cones, cylinders, and spheres.
The volume (V) of a cone is given by the formula:
V_cone = (1/3) * π * r^2 * h,
where r is the radius of the cone's base and h is its height.
The volume (V) of a cylinder is given by the formula:
V_cylinder = π * r^2 * h,
where r is the radius of the cylinder's base and h is its height.
The volume (V) of a sphere is given by the formula:
V_sphere = (4/3) * π * r^3,
where r is the radius of the sphere.
Now, let's solve the given problem.
The volume of the cylinder is given as 24x - 36 cm^3. We don't have the values for r (radius) and h (height), but we are told that the radius of the sphere is the same as the cylinder's radius and the height of the cylinder is equal to the sphere's diameter. This means that r_sph = r_cyl and h_cyl = 2 * r_sph (diameter is twice the radius).
So, we can substitute h_cyl = 2 * r_sph in the formula for the volume of a cylinder:
V_cylinder = π * r_cyl^2 * h_cyl
Substituting the values:
24x - 36 = π * r_cyl^2 * 2 * r_cyl
Simplifying:
24x - 36 = 2π * r_cyl^3
Now, we can find the volume of the sphere by substituting r_sph = r_cyl in the formula:
V_sphere = (4/3) * π * r_sph^3
Substituting the values:
V_sphere = (4/3) * π * r_cyl^3
Since the expression for the volume of the sphere is the same as the expression we obtained for 2π * r_cyl^3, we can conclude that the volume of the sphere is equal to:
V_sphere = (4/3) * (24x - 36) cm^3