6.1: Volumes of each cone are:
Radius (in.) Volume (cubic in.)
1 28.27 (rounded to nearest cubic inch)
2 113.09 (rounded to nearest cubic inch)
3 254.34 (rounded to nearest cubic inch)
6.2: There is not a linear relationship between the radius and the volume of these cones. The relationship is quadratic.
6.1: The volume of each cone and explain whether there is a linear relationship between the radius and the volume.
1) Calculating the volume of each cone:
The volume of a cone can be calculated using the following formula:
Volume = (1/3)πr²h
where:
π (pi) is a mathematical constant approximately equal to 3.14159
r is the radius of the cone's base
h is the height of the cone
In this case, the height of all the cones is 9 inches. So, we can plug in h = 9 and the different values of r to find the volume of each cone:
2) For r = 1 inch:
Volume = (1/3) × π × (1 inch)² × 9 inches = 3.14 × 1 × 9
Volume ≈ 28.27 cubic inches (rounded to the nearest cubic inch)
3) For r = 2 inches:
Volume = (1/3) × π × (2 inches)² × 9 inches = 3.14 × 4 × 9
Volume ≈ 113.09 cubic inches (rounded to the nearest cubic inch)
4) For r = 3 inches:
Volume = (1/3) × π × (3 inches)² × 9 inches = 3.14 × 9 × 9
Volume ≈ 254.34 cubic inches (rounded to the nearest cubic inch)
6.2: Is there linear relationship between radius and volume?
As you can see from calculations, the volume of the cone increases as the radius increases. However, increase is not proportional. The volume increases much faster than the radius. For example, when the radius doubles from 1 inch to 2 inches, the volume more than quadruples from 28.27 cubic inches to 113.09 cubic inches. This is because the formula for the volume includes the radius squared (r²).
