A cone has diameter 12 and height 9. A cylinder is placed inside the cone so the base of the cylinder is concentric with the base of the cone and the upper base of the cylinder is contained in the surface of the cone. If the volume of the cone is nine times the volume of the cylinder, find the dimensions of the cylinder.
Draw a side view of the figure. Using similar triangles, it is clear that
(9-h)/r = 9/6
h = 9 - 3r/2
Now we can compare volumes, and we find
1/3 pi * 6^2 * 9 = 9 pi r^2 h
108 = 9r^2(9 - 3r/2)
r^3 - 6r^2 + 8 = 0
r = 1.3054
To find the dimensions of the cylinder, we need to understand the relationship between the volumes of the cone and the cylinder.
Let's start by finding the formulas for the volume of a cone and a cylinder:
1. The volume of a cone is given by the formula: V_cone = (1/3) * π * r^2 * h, where r is the radius of the base and h is the height of the cone.
2. The volume of a cylinder is given by the formula: V_cylinder = π * r^2 * h, where r is the radius of the base and h is the height of the cylinder.
Now, let's denote the radius of the cone as R_cone and the radius of the cylinder as R_cylinder.
Since the base of the cylinder is concentric with the base of the cone, we know that the radius of the cylinder is the same as the radius of the cone: R_cylinder = R_cone.
Based on the given information, the diameter of the cone is 12 units. We can find the radius of the cone by dividing the diameter by 2: R_cone = 12 / 2 = 6 units.
The height of the cone is given as 9 units: h_cone = 9 units.
Now, let's calculate the volume of the cone: V_cone = (1/3) * π * R_cone^2 * h_cone.
Substituting the values we have: V_cone = (1/3) * π * 6^2 * 9.
Simplifying this expression: V_cone = (1/3) * π * 36 * 9 = 12 * 9 * π = 108π.
According to the problem, the volume of the cone is nine times the volume of the cylinder: V_cone = 9 * V_cylinder.
Therefore, we can set up the equation: 9 * V_cylinder = 108π.
Dividing both sides of the equation by 9, we get: V_cylinder = 108π / 9 = 12π.
To find the dimensions of the cylinder, we need to find the height of the cylinder. Let's denote it as h_cylinder.
Using the formula for the volume of a cylinder, which is V_cylinder = π * R_cylinder^2 * h_cylinder, we can substitute the known values: 12π = π * R_cylinder^2 * h_cylinder.
Since R_cylinder = R_cone = 6 and cancelling out π, we have: 12 = 6^2 * h_cylinder.
Simplifying further, we get: 12 = 36 * h_cylinder.
Dividing both sides of the equation by 36, we find: h_cylinder = 12 / 36 = 1/3.
Therefore, the height of the cylinder is 1/3 units.
In summary, the dimensions of the cylinder are:
- Radius (R_cylinder) = 6 units.
- Height (h_cylinder) = 1/3 units.