The manager of the theater wants to sell the popcorn in a cone with a diameter of 7 inches. What does the height of the cone need to be in order to hold the same amount of popcorn as the cylinder?
To determine the height of the cone that can hold the same amount of popcorn as the cylinder, we need to consider the volume of both shapes. The volume of a cylinder is calculated using the formula V_cylinder = π * r^2 * h, where r is the radius of the cylinder base and h is the height of the cylinder.
Given that the diameter of the cone is 7 inches, we can calculate the radius by dividing the diameter by 2: r_cone = 7 / 2 = 3.5 inches.
The volume of the cone is given by the formula V_cone = (1/3) * π * r^2 * h_cone, where r_cone is the radius of the cone base and h_cone is the height of the cone.
Since we want the volume of the cone to be equal to the volume of the cylinder, we can set up the following equation:
(1/3) * π * r^2 * h_cone = π * r^2 * h_cylinder
We can cancel out the common terms π * r^2 from both sides of the equation, resulting in:
(1/3) * h_cone = h_cylinder
To find the height of the cone, we can rearrange the equation to solve for h_cone:
h_cone = 3 * h_cylinder
Therefore, the height of the cone needs to be three times the height of the cylinder to hold the same amount of popcorn.
assuming the cylinder also has diameter 7, then if the cone has height h and the cylinder has height k, you need
π/3 * r^2 h = π * r^2 k
h/3 = k
The cone needs to be 3 times as tall as the cylinder