Question

Part C The volume of the pyramid is one-third of the area of its base multiplied by its height, which is   cubic units. Using this measurement and your answer from part B, derive a formula for the volume of a cone.

Answer 1

To derive the formula for the volume of a cone, we can use the information given about the volume of a pyramid and the relationship between a cone and a pyramid.

Step-by-step explanation:

To derive the formula for the volume of a cone, we can use the information given about the volume of a pyramid and the relationship between a cone and a pyramid. The volume of a pyramid is one-third of the area of its base multiplied by its height.

Since a cone can be thought of as a pyramid with a circular base, we can rewrite the formula as:

Vcone = (1/3) * Abase * hcone

Where Abase is the area of the circular base of the cone and hcone is the height of the cone.

Answer 2

To derive a formula for the volume of a cone, we can start with the formula for the volume of a pyramid and make a substitution.

The formula for the volume of a pyramid is:

Volume = (1/3) * base area * height

In this case, the volume of the pyramid is given as c cubic units, and the base area and height are not specified.

From part B, we know that the area of the base of the pyramid is equal to the area of the base of a cone, which is given by the formula:

Base Area = π * r^2

Now, let's make the substitution in the volume formula:

c = (1/3) * (π * r^2) * height

To derive the formula for the volume of a cone, we need to solve for the volume V, which is the same as the volume of the pyramid:

V = c = (1/3) * (π * r^2) * height

Simplifying the equation, we get:

V = (1/3) * π * r^2 * height

Therefore, the derived formula for the volume of a cone is:

Volume = (1/3) * π * r^2 * height