Question

Inside a right circular cone with base radius 5 and height 12 are three congruent spheres each with radius r. Each sphere is tangent to the other two spheres and also tangent to the base and side of the cone. What is r

Answer 1

The value of the radius, r is 2.24cm

Consider Triangle AEO (with AE being the slant height of the cone and EO the radius of the sphere).

This triangle is similar to the right triangle AOC (with AO being the height of the cone and OC the radius of the base). The similarity ratio is r/r + 5 (r for EO and r + 5 for OC).

Since triangles AEO and AOC are similar, we can write a proportion between corresponding sides:

(r + 5) / 12 = 1 / r

Rearrange the equation to get:

`r^2 + 5r - 12 = 0`

This is a quadratic equation that can be factored or solved using the quadratic formula.

Using the quadratic formula:

r = (-b ± √(b² - 4ac)) / 2a

where a = 1, b = 5, and c = -12.

Substitute the values

r = 2.24 cm