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
18 cm cubed
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The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height. Given that the volume of the cylinder is 36 cm^3, we can set up the equation as 36 = πr^2h.
Since the height of the cylinder is equal to the sphere's diameter, we have h = 2r. Substituting this into the equation, we get 36 = πr^2(2r), so 36 = 2πr^3.
Solving for r, we get r^3 = 18/π, and thus r = (18/π)^(1/3).
The volume of a sphere is given by the formula V = (4/3)πr^3. Substituting the value of r we found earlier, the volume of the sphere is V = (4/3)π((18/π)^(1/3))^3 = 24 cm^3.
Therefore, the volume of the sphere is 24 cm^3.
The correct response is:
24 cm3
24 cm cubed