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?
To find the radius of the congruent spheres, we need to use the Pythagorean theorem and consider the properties of the right circular cone.
Let's consider the cross-section of the cone through its apex, which creates a triangle. The radius of the base of the cone is given as 5, and the height of the cone is given as 12.
Each sphere is tangent to the base of the cone. Therefore, the radius of the sphere, r, is equal to the perpendicular distance from the center of the base to the side of the cone, which we'll call x.
Using the properties of similar triangles in the cone triangle, we can set up the following ratios:
r / x = (r + r) / (x + 5)
Simplifying this equation, we get:
r / x = 2r / (x + 5)
Cross-multiplying, we have:
r(x + 5) = 2rx
Expanding this equation yields:
rx + 5r = 2rx
Now let's solve for x:
5r = rx
Dividing both sides by r:
5 = x
Finally, we can substitute this value of x back into either of the original equations to find r:
r / 5 = (r + r) / (5 + 5)
r / 5 = 2r / 10
Cross-multiplying:
10r = 10r
Dividing both sides by 10 gives:
r = r
Since r is the radius of the sphere, the value of r can be any positive real number. So, the radius of the congruent spheres can be any positive real number.