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)
Responses
24 cm3
24 cm cubed
54 cm3
54 cm cubed
12 cm3
12 cm cubed
18 cm3
A B C D
We can use the relationship between the volume of a cylinder and the volume of a cone to solve this problem.
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height.
Since the height of the cylinder is equal to the diameter of the sphere, we can say that h = 2r.
Substituting this value into the volume formula for the cylinder, we have V = πr^2(2r) = 2πr^3.
The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius.
Since the radius of the sphere is the same as the cylinder's, we can say that V = (4/3)πr^3 = 2πr^3.
From the given information, we know that the volume of the cylinder is 36 cm^3.
Setting 2πr^3 equal to 36, we can solve for r.
2πr^3 = 36
r^3 = 36/(2π)
r^3 = 18/π
r = (18/π)^(1/3)
Substituting this value of r into the volume formula for the sphere, we have
V = (4/3)π[(18/π)^(1/3)]^3
Simplifying this expression, we get
V = (4/3)π(18/π)
V = (4/3)(18)
V = 24 cm^3
Therefore, the volume of the sphere is 24 cm^3.
The correct answer is A. 24 cm^3.