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
24 cm3
24 cm cubed
54 cm3
54 cm cubed
12 cm3
12 cm cubed
The volume of a cylinder is given by the formula Vcylinder = πr^2h.
The height of the cylinder is equal to the sphere's diameter, so h = 2r.
Substituting this value into the volume formula, we get Vcylinder = πr^2 * 2r = 2πr^3.
We are given that the volume of the cylinder is 36 cm^3, so 2πr^3 = 36.
To find the volume of the sphere, we can use the formula Vsphere = (4/3)πr^3.
Substituting the value of r from the equation above, we get Vsphere = (4/3)π * (36/(2π)) = (4/3) * 18 = 24 cm^3.
Therefore, the volume of the sphere is 24 cm^3.
The volume of a cone is 253 π cm3 . What is the volume of a sphere if its radius is the same as the cone’s and the height of the cone is equal to the sphere’s diameter?(1 point)
Responses
252 π cm3
Start Fraction 25 over 2 End Fraction pi cm cubed
503 π cm3
Start Fraction 50 over 3 End Fraction pi cm cubed
256 π cm3
Start Fraction 25 over 6 End Fraction pi cm cubed
25 π cm3
25 pi cm cubed
The volume of a cone is given by the formula Vcone = (1/3)πr^2h.
The height of the cone is equal to the sphere's diameter, so h = 2r.
Substituting this value into the volume formula, we get Vcone = (1/3)πr^2 * 2r = (2/3)πr^3.
We are given that the volume of the cone is 253π cm^3, so (2/3)πr^3 = 253π.
To find the volume of the sphere, we can use the formula Vsphere = (4/3)πr^3.
Substituting the value of r from the equation above, we get Vsphere = (4/3)π * (253π/(2/3)π) = (4/3) * (253/2) = 507/3 = 169.
Therefore, the volume of the sphere is 169 cm^3.