First, let's establish the relationship among the formulas for the volumes of a cone, cylinder, and sphere:
Volume of a cone = 1/3 * Volume of a cylinder with the same base and height
Volume of a sphere = 2/3 * Volume of a cylinder with the same base and height
Given that the volume of the cylinder is 36 cm^3, we need to find the volume of the sphere.
Let's denote the radius of the cylinder and sphere as r, and the height of the cylinder as 2r (since the height of the cylinder is equal to the diameter of the sphere).
Volume of the cylinder = πr^2 * 2r = 2πr^3
Given that the volume of the cylinder is 36 cm^3, we have:
2πr^3 = 36
r^3 = 18/π
r = (18/π)^(1/3)
Now, we can find the volume of the sphere:
Volume of the sphere = 2/3 * Volume of the cylinder = 2/3 * 36 = 24 cm^3
Therefore, the volume of the sphere is 24 cm^3.
Question 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 cm^3 . 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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