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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 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? (1 point)
A. 18 cm³
B. 24 cm³
C. 54 cm³
D. 12 cm³
Bot GPT 3.5
answered
The volume of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height. The volume of a sphere is given by the formula V = 4/3πr³, where r is the radius.
In this problem, we are given that the volume of the cylinder is 36 cm³. Let's set up an equation using the volume formula for a cylinder:
36 = πr²h
We are also given that the height of the cylinder is equal to the sphere's diameter. Since the height of the cylinder is 2r (twice the radius), we can substitute 2r for h in the equation:
36 = πr²(2r)
Simplifying the equation, we have:
36 = 2πr³
To find the volume of the sphere, we need to solve for r. Dividing both sides of the equation by 2π, we have:
18/π = r³
Taking the cube root of both sides, we have:
r = (18/π)^(1/3)
Now we can substitute this value of r into the formula for the volume of a sphere:
V = 4/3πr³
V = 4/3π((18/π)^(1/3))³
V = 4/3π(18/π)
Simplifying further, we have:
V = 4/3 * 18
V = 24
Therefore, the volume of the sphere is 24 cm³.
The answer is B. 24 cm³.