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 256 π cm3 Start Fraction 25 over 6 End Fraction pi cm cubed 25 π cm3 25 pi cm cubed 503 π cm3
Let's first find the dimensions of the cone in terms of the volume:
Vcone = 1/3 * π * r^2 * h
253π = 1/3 * π * r^2 * h
Since the height of the cone is equal to the sphere's diameter, we can also write h = 2r:
253π = 1/3 * π * r^2 * 2r
253π = 2/3 * π * r^3
253π = 2π/3 * r^3
r^3 = (253π * 3) / 2
r^3 = 379.5
r = ∛379.5
r ≈ 7.64 cm
Now, let's calculate the volume of the sphere with a radius of 7.64 cm:
Vsphere = 4/3 * π * r^3
Vsphere = 4/3 * π * (7.64)^3
Vsphere = 4/3 * π * 440.55
Vsphere ≈ 1849.57 π cm^3
Therefore, the volume of the sphere is approximately 1849.57 π cm^3, which is closest to 252 π cm^3.
Therefore, the correct response is 252 π cm^3.
A sculptor created a design by carving a cone out of a cylinder. The cone and cylinder share the same radius and height. If the volume of the cylinder before removing the cone is 54 in.3 , what is the volume of the amount remaining?(1 point) Responses 36 in.3 36 in. cubed 27 in.3 27 in. cubed 40 in.3 40 in. cubed 18 in.3
The total volume of the cylinder before the cone is removed would be the volume of the cylinder itself. We are given that the volume of the cylinder is 54 in^3.
Now, the volume of a cylinder is given by:
V_cylinder = π * r^2 * h
Since the cone is carved out of the cylinder, the remaining volume when the cone is removed would be:
V_remain = V_cylinder - V_cone
Given that the cone and cylinder share the same radius and height, the volume of the cone would be:
V_cone = 1/3 * π * r^2 * h
Substituting the formula for the volumes:
V_remain = 54 - 1/3 * π * r^2 * h
Since the cone and cylinder share the same radius and height, the radius (r) and height (h) will cancel out in the subtraction leaving us with:
V_remain = 54 - 1/3 * π * r^2 * h
V_remain = 54 - 54/3
V_remain = 54 - 18
V_remain = 36 in^3
Therefore, the volume of the amount remaining after carving the cone out of the cylinder is 36 in^3.
The correct response is 36 in^3.