In mathematical terms, I've got a problem where I'd like to calculate the volume of a cylinder whose upper surface is bounded by a right circular cone. A brief discription of the geometries follow, but it's probably best to look at the attachments.
In the real world, a description of the problem follows: consider the cylinder an inground tank with radius 7.5 m and height 8.1 m. The tank is used to collect sand form the top edge [along the edge of the tank] such that the sand entering the tank creates the shape of half of a right circular cone of radius 14 m [and height 8.1 m]. Notice that the cone's base does not touch the far edge of the tank [in fact, it is 1 m away from the edge].
What is the volume of sand in the tank?
1. I decided to use a triple integral in cylindrical coordinates.
2. I defined my cylinder equation
3. I defined my cone equation
4. I identified the angles to integrate through
5. Compute the triple integral
But my answer is obviously wrong since the volume that I'm calculating is larger than the total volume of the tank.
Can anyone offer help? I've checked and re-checked. What am I not seeing? Is my method wrong? Or is my arithmetic wrong? Any help would be greatly appreciated!!
Also, I would have loved to type out the question in the thread -- but sadly, i dont know how to.
Thanks in advance!