First, we need to find the radius and height of the cylinder. We know that the volume of a cylinder is given by the formula V_cylinder = πr^2h, where r is the radius and h is the height. We are given that V_cylinder = 36 m³.
Using the formula for the volume of a cylinder, we can rearrange it to solve for h:
V_cylinder = πr^2h
36 = πr^2h
h = 36 / (πr^2)
We are also given that the height of the cylinder is equal to the diameter of the sphere, which means h = 2r.
Substituting h = 2r into the equation above, we get:
2r = 36 / (πr^2)
2r^3 = 36 / π
r^3 = 18 / π
r = (18 / π)^(1/3)
Now that we have found the radius of the cylinder, we can use the formula for the volume of a sphere V_sphere = 4/3 * πr^3 to find the volume of the sphere:
V_sphere = 4/3 * π * [(18 / π)^(1/3)]^3
V_sphere = 4/3 * π * 18
V_sphere = 24π
Therefore, the volume of the sphere is 24π m³.
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 36 m³. 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?
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