A cylinder and a cone have the same diameter: 8 inches. The height of the cylinder is 3 inches. The height of the cone is 18 inches.
Use π = 3.14.
What is the relationship between the volume of this cylinder and this cone? Explain your answer by determining the volume of each and comparing them. Show all your work. (10 points)
To determine the relationship between the volume of the cylinder and the cone, we first need to calculate the volume of each.
The volume of a cylinder can be calculated using the formula:
V_cylinder = π * r^2 * h_cylinder
where r is the radius of the cylinder and h_cylinder is the height of the cylinder.
Given that the diameter of the cylinder is 8 inches, the radius (r) is half of that:
r = diameter / 2
r = 8 / 2
r = 4 inches
And the height of the cylinder is 3 inches.
V_cylinder = π * 4^2 * 3
V_cylinder = 3.14 * 16 * 3
V_cylinder = 150.72 cubic inches
The volume of a cone can be calculated using the formula:
V_cone = (1/3) * π * r^2 * h_cone
where r is the radius of the cone and h_cone is the height of the cone.
Since the diameter of the cone is also 8 inches, the radius (r) is again half of that:
r = diameter / 2
r = 8 / 2
r = 4 inches
And the height of the cone is 18 inches.
V_cone = (1/3) * 3.14 * 4^2 * 18
V_cone = (1/3) * 3.14 * 16 * 18
V_cone = 301.44 cubic inches
Now, let's compare the volumes of the cylinder and the cone:
V_cylinder = 150.72 cubic inches
V_cone = 301.44 cubic inches
Since the volume of the cone is approximately two times the volume of the cylinder, we can conclude that the volume of the cone is twice the volume of the cylinder.