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University of Southern California

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Problem 86

A right circular cone has base of radius 1 and height 3.A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side-length of the cube?

Answer

$\frac{3 \sqrt{2}}{\sqrt{2}+3}(\approx 0.9611)$

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## Discussion

## Video Transcript

we're giving kind of challenging problems. Let's go ahead and walk this together. So we're told that a right circular cone has a base radius of one based radius of one and a height of three. Okay. Of cube is inscribed in the cones. About one face of the cube is contained in the base of the cone. What is the side length of that cube? Okay, well, we need to consider the cross section obtained by slicing vertically by a plane that contains diagonal off the base of the cube so we can obtain the rectangle of height. We'll just call that high s. So the height s side length of the cube and the with which in the note that as rad two times s rad two of us. Um and we could do this by using similar triangles. So we go ahead and we say, All right, so we have ah, height of three in a base of one, and that is equal to s all over. One minus 1/2 off. And then we take it this bad boy equation right here, the, um the square root of two times s perfect. So now we can go ahead and say that as is equal to six over three, Red two plus two. If we go ahead and we simplify this or we're basically we're first going to expand this to get a light denominator so we can say six times three rod to minus two all over three Rad two plus two times three rat to minus two. That is equal to nine. Rad to minus six all over seven. And that is our answer. That is our answer. Who want to go ahead and we want to go ahead and now evaluate this. We can say this is approximately 0.9611