a ball of radius 1 is dropped into a V-shaped gutter. The ball touches the gutter's sides at the ball's equator. If the shortest distance from the surface of the ball to the vertex of the gutter is 1, what is the angle formed by the sides of the gutter?
To solve this problem, we can use the concept of similar triangles.
Let's assume the vertex of the gutter is point A, and the points where the ball touches the gutter are B and C. Let the center of the ball be point O, and let D be the point where the ball's equator touches the gutter.
First, let's draw a line segment from point O to point D, and another line segment from point O to point A. We can see that triangle OAD is a right triangle, since it has a vertex at point A, and line segment OD is the shortest distance from the surface of the ball to the vertex of the gutter (which is given to be 1).
Now, let's consider triangle OBD. Since the ball is tangent to the gutter's sides at the ball's equator, line segment BD is perpendicular to line segment OA. Since line segment OB is the radius of the ball (which is given to be 1), triangle OBD is also a right triangle.
Since triangle OAD and triangle OBD are both right triangles, they share an angle at point O. Let's call this angle theta.
Now we can observe that triangle OAD and triangle OBD are similar triangles. This is because they have a shared angle (at point O) and both have a right angle.
Using the concept of similar triangles, we can set up the following proportion:
(AD / OA) = (BD / OB)
If we substitute the given values into this proportion, we have:
(1 / OA) = (BD / 1)
Simplifying this equation, we can express BD in terms of OA:
BD = OA
Since BD is the shortest distance from the surface of the ball to the vertex of the gutter (which is given to be 1), we have:
BD = 1
Substituting this value back into our proportion, we have:
1 = OA
So, OA = 1.
Now we know that OA = 1, which means triangle OAD is a 1-1-sqrt(2) right triangle. Therefore, angle OAD is equal to 45 degrees.
Since angle OAD is equal to 45 degrees and line segment OB is the radius of the ball (which is given to be 1), angle BOD is also 45 degrees (since triangle OBD is a right triangle).
Finally, the angle formed by the sides of the gutter, which is angle BOC, is equal to 90 minus (45 plus 45), which is equal to 90 - 90, which equals 0 degrees.
Therefore, the angle formed by the sides of the gutter is 0 degrees.