Question

Suppose a sphear, cylinder, and cone all share the same radius and the cylinder and cone also share the same height, which is twice the radius. which of the following is true about the relationship between the volume formulas for the sphear, cylinder, and cone?

Answer 1

The relationship between the volume formulas for the sphere, cylinder, and cone is as follows:

The volume of a sphere is given by the formula: V(sphere) = (4/3)πr^3.

The volume of a cylinder is given by the formula: V(cylinder) = πr^2h.

The volume of a cone is given by the formula: V(cone) = (1/3)πr^2h.

In this case, since the radius is the same for all three, and the height of the cylinder and cone is twice the radius, we can substitute this value into the formulas.

For the sphere, the volume would be: V(sphere) = (4/3)πr^3.

For the cylinder, the volume would be: V(cylinder) = πr^2(2r) = 2πr^3.

For the cone, the volume would be: V(cone) = (1/3)πr^2(2r) = (2/3)πr^3.

Thus, we can see that the relationship between the volume formulas for the sphere, cylinder, and cone is:

V(sphere) : V(cylinder) : V(cone) = (4/3)πr^3 : 2πr^3 : (2/3)πr^3.

Simplifying, this becomes:

V(sphere) : V(cylinder) : V(cone) = 4 : 6 : 2.

Therefore, the true relationship is that the volume of the sphere is one-third the volume of the cylinder, and the volume of the cone is one-sixth the volume of the cylinder.