Question

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?

Short answer

The side length of the cube is \(sqrt(2)\).

Step-by-step solution

Understand the relationship

The first step is recognizing that the cube’s face diagonal is the same as the diameter of the cone base, and the cube’s height is the same as the radius of the cone base. This is because the cube is inscribed in the cone.

Set up an equation

The next step is to set up an equation, which is based on the properties of the cube. The face diagonal ('d') of a cube of side 'a' can be found using the Pythagorean theorem \(a^{2} + a^{2} = d^{2}\). Thus \(2a^{2} = d^{2}\), and d = \(sqrt(2) * a\). In this case, d is equal to two times the radius of the base of the cone, so we can write the equation as \(sqrt(2) * a = 2\*1\).

Solve for side-length 'a'

With our equation in place, the final step is to solve for 'a'. Rearranging the equation to solve for 'a', we get \(a = \frac{2}{sqrt(2)} = sqrt(2)\).