Use a relationship I’m on the formulas of the volumes of cones, cylinders and spheres to solve the following problem. The volume of the cylinder is 36 cm to the third power. What is the volume of the sphere if it’s radius is the same amount of the cylinders in the height of the cylinder is equal to the spheres diameter.
Let's assume the radius of the cylinder is r cm and the height of the cylinder is also r cm.
The volume of a cylinder is given by the formula: V_cylinder = πr^2h.
Given that the volume of the cylinder is 36 cm^3, we can write the equation as:
36 = πr^2 * r.
Simplifying this equation, we have:
36 = πr^3.
Now, let's find the volume of the sphere using the relationship between the cylinder and sphere volumes. The volume of a sphere is given by the formula: V_sphere = (4/3)πr^3.
Since the radius of the cylinder is the same as the radius of the sphere, and the height of the cylinder is equal to the diameter of the sphere, we can substitute r for the radius and 2r for the height in the cylinder volume formula:
36 = πr^3.
From this equation, we can see that r^3 = 36/π.
Now let's substitute the value of r^3 into the sphere volume formula:
V_sphere = (4/3)πr^3.
V_sphere = (4/3)π * (36/π).
V_sphere = 4 * (36/3).
V_sphere = 4 * 12.
V_sphere = 48 cm^3.
Therefore, the volume of the sphere is 48 cm^3.