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 find the angle formed by the sides of the gutter, we can start by visualizing the situation described. First, let's picture the V-shaped gutter and the ball dropped into it.

The ball is dropped in such a way that it touches the sides of the gutter at its equator, which means the ball will be perfectly centered within the gutter.

The shortest distance from the surface of the ball to the vertex of the gutter is given as 1. Since the ball has a radius of 1, this implies that the vertex of the gutter is located at the center of the ball.

Now, let's consider the situation from a top-down or planar view, looking directly into the gutter. In this view, we can see that the gutter forms an isosceles triangle with the vertex of the gutter and the points where the ball touches the sides.

Since the ball touches the sides of the gutter at its equator, the distance from the vertex of the gutter to the equator of the ball is also equal to 1.

We can form right triangles by connecting the vertex of the gutter to each of the points where the ball touches the sides. Since the distance from the vertex to the ball's surface is given as 1, and the radius of the ball is also 1, we can infer that each of these right triangles has legs of length 1.

Furthermore, since the ball is perfectly centered within the gutter, the triangle formed by the vertex and the points where the ball touches the sides is isosceles. This means that the two legs of the right triangles, which are both of length 1, are congruent.

To determine the angle formed by the sides of the gutter, we need to find the measure of one of the angles within the isosceles triangle. Let's call this angle x.

In a triangle, the sum of the interior angles is always 180 degrees. Therefore, we can write the equation:

2x + 90 = 180

Simplifying this equation, we get:

2x = 90

Dividing both sides by 2, we find:

x = 45

Therefore, the angle formed by the sides of the gutter is 45 degrees.