Use the relationships 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?
a 12 cm^3
b 24 cm^3
c 18 cm^3
d 54 cm^3
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.
Since the volume of the cylinder is given as 36 cm^3, we can set up the equation 36 = πr^2h.
We are given that the height of the cylinder is equal to the sphere's diameter, which means h = 2r.
Substituting this into the equation, we have 36 = πr^2(2r).
Simplifying further, 36 = 2πr^3.
Dividing both sides by 2π, we get r^3 = 18.
Taking the cube root of both sides, we find that r = 3.
Now, we can use the volume formula for a sphere, which is V = (4/3)πr^3.
Substituting the radius we found earlier, we have V = (4/3)π(3)^3 = 36π.
The volume of the sphere is therefore 36π cm^3. Since π is approximately 3.14, the volume is approximately 36(3.14) = 113.04 cm^3.
None of the listed answer choices match this calculated value.