### Brain Teasers

# Food Fight Fiasco #1

You are in a food fight. An enemy is holding a hot sauce and will spray you with the fiery substance unless you answer the following question correctly:

A right cylinder, with a diameter congruent to its height is inscribed in a right cone with height 12 units and radius 5 units. If their axes coincide, find the radius of the cylinder.

A right cylinder, with a diameter congruent to its height is inscribed in a right cone with height 12 units and radius 5 units. If their axes coincide, find the radius of the cylinder.

### Hint

Hint #1: By inscribed, your enemy means that the cylinder's whole bottom face is touching the cone's bottom face, and the cylinder is touching the cone with the circumference of the top face.Hint #2: There are many ways to solve this, but a simple method is to imagine one triangle inside another.

### Answer

30/11. As mentioned in the hint, there are many methods in which to solve this. One less complex method is to imagine the axis of the cone form a right triangle. This triangle shares the dimensions of the cone, so it is 12 by 5. We are given that the diameter of the cylinder is congruent to the height, so we can deduce that the radius is half the length of the height. Imagine a radius that intersects the hypotenuse of the triangle discussed above. Let's call it x. Therefore, we get a right triangle with a smaller right triangle inside. Since two of the angles are the same (the top and the right angles), we can say that the two triangles are similar. Since the height of the cylinder is 2x (twice the length of the radius x), the height of the smaller right triangle is 12-2x. Thus, we can set up a proportion: 12/5 = 12-2x/x.Next we can cross-multiply to get 12x=60-10x. If we add 10x on both sides, we get 22x=60, and dividing 22 from both sides yields x=60/22=30/11.

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