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.
Make a diagram showing the cross-section.
let the radius of the cylinder be r and its height be h
The small triangle to the right of the cylinder is similar to the triangle formed within the cone
so h/(6-r) = 9/6 = 3/2
2h = 18 - 3r, where r < 6
h = (18-3r)/2
volume of cone = (1/3)?(36)(9) = 108? cubic units
volume of cylinder = ?(r^2)(h)
= ?(r^2)(18-3r)/2
108? = 9(?(r^2)(18-3r)/2)
216? = 162?r^2 - 27?r^3
divide by 27?
8 = 6r^2 - r^3
r^3 - 6r^2 + 8 = 0
Here comes the hard part,
This equation does not factor, I have no idea if you know how to solve a cubic. One method is Newton's Method.
Another is to use a webbased method like Wolfram
http://www.wolframalpha.com/input/?i=r%5E3+-+6r%5E2+%2B+8+%3D+0
we get 2 positive solutions for
r = 1.3054 , then h = 7.0419
r = 5.7588 , then h = .3618
both solutions are valid
To solve this problem, we need to find the dimensions of the cylinder.
Let's start by finding the volume of the cone. The formula for the volume of a cone is given by V_cone = (1/3) * π * r^2 * h, where r is the radius of the base and h is the height of the cone.
Given that the diameter of the cone is 12, we can find the radius by dividing the diameter by 2: r_cone = 12/2 = 6.
Substituting the values into the formula, we have V_cone = (1/3) * π * (6^2) * 9 = 108π.
Now, let's find the volume of the cylinder. The formula for the volume of a cylinder is given by V_cylinder = π * r^2 * h_cylinder, where r is the radius of the base and h_cylinder is the height of the cylinder.
Since the base of the cylinder is concentric with the base of the cone, they have the same radius, r_cylinder = r_cone = 6.
Let's assume the height of the cylinder is h_cylinder.
According to the problem, the volume of the cone is nine times the volume of the cylinder: V_cone = 9 * V_cylinder.
Substituting the volumes we found earlier, we have 108π = 9 * (π * (6^2) * h_cylinder).
Simplifying the equation, we can cancel out π and solve for h_cylinder:
108 = 9 * 36 * h_cylinder
108 = 324 * h_cylinder
h_cylinder = 108 / 324
h_cylinder = 1/3
Therefore, the height of the cylinder is 1/3.
To summarize, the dimensions of the cylinder are a radius of 6 (which is the same as the cone), and a height of 1/3.