Each row of the table below represents a cone with a height of 9 inches and a different radius.

6.1

Calculate the volume of each cone.

Write your answer in terms of it or rounded to the nearest cubic inch.

Radius (in.)

Volume (cubic in.)

1

2

3

9 in.

6.2

Is there a linear relationship between the radius and the volume of these cones?

Explain your thinking. no

The volume of cones with radii of 1, 2, and 3 inches and a height of 9 inches are approximately 9, 38, and 85 cubic inches, respectively, showing a cubic relationship between the radius and the volume, not linear.

Explanation:

The volume of a cone is given by the formula V = (1/3)$\pi$r^2h, where V is the volume, r is the radius of the base, and h is the height of the cone. Given that the height of each cone is 9 inches, we can calculate the volume for cones with radii 1 inch, 2 inches, and 3 inches.

For r = 1 inch: V = (1/3)$\pi$(1)^2(9) = 3$\pi$ cubic inches or approximately 9 cubic inches.
For r = 2 inches: V = (1/3)$\pi$(2)^2(9) = 12$\pi$ cubic inches or approximately 38 cubic inches.
For r = 3 inches: V = (1/3)$\pi$(3)^2(9) = 27$\pi$ cubic inches or approximately 85 cubic inches.

Regarding the relationship between the radius and the volume of these cones, it is not linear. The volume of a cone changes with the cube of its radius, indicating a cubic relationship (V \propto r^3), as demonstrated by the formula for volume.