SOLUTION: A block of wood is in the form of a right circular cone. The altitude is 12 cm and the radius of the base is 5 cm. A cylindrical hole 5 cm in diameter is bored completely through t
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Question 1198596: A block of wood is in the form of a right circular cone. The altitude is 12 cm and the radius of the base is 5 cm. A cylindrical hole 5 cm in diameter is bored completely through the solid, the axis of the hole coinciding with the axis of the cone. The amount of wood left after the hole is bored may be expressed as V= xπ/(γ ) 〖cm〗^3 where x is a positive integer and y is a prime number. Find ∛x + γ.
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!
The radius of the base of the cone is 5cm, and the radius of the hole that is bored through the cone is 2.5cm. Similar triangles then tell us that the heights of the cylindrical part of the hole and the conical part of the hole are both 6cm.
The volume of wood left after the hole is bored is the volume of the original cone, minus the volumes of the conical and cylindrical parts of the hole.
volume of original cone: (1/3)(pi)(5^2)(12)=100pi
volume of conical part of hole: (1/3)(pi)(2.5^2)(6)=12.5pi
volume of cylindrical part of hole: (pi)(2.5^2)(6)=37.5pi
Volume of wood remaining: 100pi - (1.25pi+37.5pi) = 50pi
Another path to that answer, without using the actual volume formulas, knowing that the heights of the conical and cylindrical parts of the hole are equal:
The height of the conical part of the hole is half the height of the original cone, so the volume of the conical part of the hole is 1/8 of the volume of the original cone.
The volume of the cylindrical part of the hole, having the same radius as the conical part of the hole, is 3 times the volume of the conical part of the hole, which is 3/8 the volume of the original cone.
So the total volume of the hole is (1/8)+(3/8) = 1/2 the volume of the original cone. And of course that means the volume of the solid remaining after the hole is drilled is 1/2 the volume of the original cone.
The volume of the original cone with base radius 5 and height 12 is 100pi; so the volume of the solid after the hole is drilled is 50pi.
But now there is a problem: The answer of 50pi cannot be expressed in the specified form "x(pi)/y where x is a positive integer and y is a prime number".
So the problem is apparently posted incorrectly....
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