It took me a while to figure this out, but here goes.
The Problem was:
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?
The two images linked below show the solution.
The first is a top view, looking down on the base of the cone.
The second is a side view of one of the spheres, showing how it touches the base and sides of the cone.
ibb.co/tJpkBWt
ibb.co/7kZYNsj
That was fun.
No images. Do not want to follow links, because of the danger of spam.
186
I'm sorry, but I need some context to understand what "186" refers to. Can you please provide more information or clarify your question?
To solve this problem, we can use the properties of similar triangles and the Pythagorean theorem.
Let's start by considering the top view of the cone. In the top view, we can see three spheres tangent to each other and tangent to the base of the cone. Since the spheres are congruent, the centers of the spheres form an equilateral triangle.
Now, let's focus on one of the spheres and its side view. From the side view, we can see that the sphere is tangent to the side of the cone and also tangent to the base.
Using the Pythagorean theorem, we can form a right triangle with the height of the cone (12), half the base of the equilateral triangle (r), and the radius of the sphere (r). The hypotenuse of this right triangle is the distance from the center of the sphere to the top of the cone.
Applying the Pythagorean theorem, we have:
(r^2) + (r^2) = (12^2) - (r^2)
Simplifying this equation, we get:
2r^2 = 144 - r^2
Combining like terms, we have:
3r^2 = 144
Dividing both sides by 3, we get:
r^2 = 48
Taking the square root of both sides, we find:
r = √48 = 4√3
Therefore, the value of r is 4√3.