Mock AIME 1 Pre 2005 Problems/Problem 8
, a rectangle with and , is the base of pyramid , which has a height of . A plane parallel to is passed through , dividing into a frustum and a smaller pyramid . Let denote the center of the circumsphere of , and let denote the apex of . If the volume of is eight times that of , then the value of can be expressed as , where and are relatively prime positive integers. Compute the value of .
As we are dealing with volumes, the ratio of the volume of to is the cube of the ratio of the height of to .
Thus, the height of is times the height of , and thus the height of each is .
Thus, the top of the frustum is a rectangle with and .
Now, consider the plane that contains diagonal as well as the altitude of . Taking the cross section of the frustum along this plane gives the trapezoid , inscribed in an equatorial circular section of the sphere. It suffices to consider this circle.
First, we want the length of . This is given by the Pythagorean Theorem over triangle to be . Thus, . Since the height of this trapezoid is , and extends a distance of on either direction of , we can use a 5-12-13 triangle to determine that .
Now, we wish to find a point equidistant from , , and . By symmetry, this point, namely , must lie on the perpendicular bisector of . Let be units from in . By the Pythagorean Theorem twice, Subtracting gives . Thus and .
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