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
18 cm3
18 cm cubed
54 cm3
54 cm cubed
12 cm3
12 cm cubed
24 cm3
54 cm3
Since the volume of a cylinder is given as 36 cm3 and the height of the cylinder is equal to the sphere's diameter, we can use the formula V_cylinder = πr^2h, where r is the radius and h is the height.
Given that the radius of the cylinder is the same as the sphere's, we can say that r_cylinder = r_sphere. Since the height of the cylinder is equal to the sphere's diameter, the height of the cylinder is 2r.
Now, we can plug these values into the cylinder volume formula:
36 = πr^2(2r)
36 = 2πr^3
r^3 = 18/π
r = (18/π)^(1/3)
Now, we can use the formula for the volume of a sphere V_sphere = (4/3)πr^3, substituting the r value we found previously:
V_sphere = (4/3)π((18/π)^(1/3))^3
V_sphere = (4/3)π((18/π))^1
V_sphere = (4/3)π(18/π)
V_sphere = 4(18/3)
V_sphere = 54 cm3
Therefore, the volume of the sphere is 54 cm3.