Geometry
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 crosssection.
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/(6r) = 9/6 = 3/2
2h = 18  3r, where r < 6
h = (183r)/2
volume of cone = (1/3)?(36)(9) = 108? cubic units
volume of cylinder = ?(r^2)(h)
= ?(r^2)(183r)/2
108? = 9(?(r^2)(183r)/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
posted by Reiny
Respond to this Question
Similar Questions

Geometry
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 
8th grade math
A cylinder has a cone on top. The base of the cone is congruent to the base of the cylinder. The diameter of the base is 16ft. The height of the cylinder is 16ft. and the height of the cone is 9th. What is the volumeof the entire 
PreAlgebra
Given a cylinder and a cone with the same base and the same height Cylinder: r=4 ft and h=3 ft Cone: r=4 ft and h=3 ft What is the volume of the cone in cubic feet? A. 12 π B. 16 π C. 36 π D. 48 π 
geometry
A cylinder has a height of 16 cm and a radius of 5 cm. A cone has a height of 12 cm and a radius of 4 cm. If the cone is placed inside the cylinder as shown, what is the volume of the air space surrounding the cone inside the 
geometry
A cylinder has a height of 16 cm and a radius of 5 cm. A cone has a height of 12 cm and a radius of 4 cm. If the cone is placed inside the cylinder as shown, what is the volume of the air space surrounding the cone inside the 
9th grade geometry
Rounded to the nearest hundred cubic meters, what is the total capacity (cone and cylinder) of the storage container? the cone is on top of the cylinder. the radius of the cone is 6 meters and the height of the cone is 8 meters. 
math
A paperweight containing liquid is made up of a cone and a cylinder. The radius of the cone is 3cm with height of 4cm. The diameter is 20cm and height is 8cm. 1. Calculate the total volume of the cone and cylinder when it si 
Calculus
Given a right circular cone, you put an upsidedown cone inside it so that its vertex is at the center of the base of the larger cone, and its base is parallel to the base of the larger cone. If you choose the upsidedown cone to 
Calculus: Optimization
I have no idea how to approach this problem, if someone knows just how to relate h, r with H,R, that would be extremely helpful and I can workout the rest! Thank you in advance. Given a right circular cone, you put an upsidedown 
Geometry
a cone full of water has been poured intoa cylinder. what is the depth of thewater in the cylinder? the cone has a height of 15m and a diameter of 20m. the cylinder also has a diameter of 20m and the amount of water in the