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Tuesday, October 16, 2012

Right Circular Cone Problem

Category: Solid Geometry

"Published in Newark, California, USA"

Two vertical conical tanks (both inverted) have their vertices connected by a short horizontal pipe. One tank, initially full of water, has an altitude of 6 feet and a diameter of base 7 feet. The other tank, initially empty, has an altitude of of 9 feet and a diameter of base 8 feet. If the water is allowed to flow through the connecting pipe, find the level to which the water will ultimately rise in the empty tank. (Neglect the water in the pipe.)

Solution:

Initial:
Photo by Math Principles in Everyday Life

Final:
Photo by Math Principles in Everyday Life

Consider the initial figure, 



 
Now, if the water is allowed to flow through the connecting pipe, then the final height of the first tank is equal to the height of the second tank after it is raised.

Consider the vertical section of the first tank at the final figure. We found out that the initial radius and height of a tank is similar to the final radius and height of a tank. Using similar triangles, we have


 
Let R1 be the final radius. Therefore,


Consider the vertical section of the second tank at the final figure. We found out that the radius and height of an empty tank is similar to the final radius and height of a tank after the water is raised. Using similar triangles, we have


 
Let R2 be the final radius. Therefore,

 
Equate equations 1 and 2, we have
 
 
 
                                            
Final Volume of the first tank,




Final Volume of the second tank,



 
Final Volume of the first tank + Final Volume of the second tank = Initial Volume of the first tank



we can cancel π as their common factor,


but 

Hence, the above equation becomes







Therefore, the final height of water is