Cube in a Cone
Date: 12/17/2002 at 18:17:10 From: Jackie Subject: Cube inside a cone I have a cone with a cube inside of it. The radius of the bottom of the cone is 1 cm.; the height of the cone is 3 cm. The cube touches the cone at the four top corners and hits somewhere toward the middle of the cone. The bottom of the cube rests directly in the center of the base of the cone but does not touch the circumference of the base. I need to know how big each side of the cube is, based on this information. I also need to be able to explain how I arrived at the answer.
Date: 12/18/2002 at 01:51:22 From: Doctor Greenie Subject: Re: Cube inside a cone Hello, Jackie - Let the length of the side of the cube be x. Then the height of the cube is x, so the distance from the top of the cube to the top of the cone is (3-x). The four vertices of the cube that touch the cone lie on a circle that is in a plane parallel to the plane containing the base of the cone; the diameter of this circle is the diagonal of the face of the cube, or x*sqrt(2). If we now look at the two-dimensional figure we get when viewing the cone directly from the side, the entire cone and the portion of the cone above the cube appear as similar triangles. From the heights and bases of these two similar triangles, we get 3-x x*sqrt(2) --- = --------- 3 2 You can solve this proportion to find the value of x. I hope this helps. Please write back if you have any further questions about any of this. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/
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