A billiard ball is struck by a cue. It travels 100 centimeters before ricocheting off a rail forming a 45 degree angle and traveling another 120 centimeters into a corner pocket. how far is the initial position of the ball from the corner pocket the ball was sunk in?Round answer to the nearest centimeter.

To find the distance between the initial position of the ball and the corner pocket, we can divide this problem into two right triangles: one formed by the initial path of 100 centimeters and the angle of reflection, and another formed by the path of 120 centimeters and the angle of reflection.

Let's start with the first triangle formed by the initial path of 100 centimeters and the angle of reflection. We know that the angle formed is 45 degrees. Using trigonometric functions, we can find the horizontal and vertical components of this triangle.

The horizontal component of the first triangle can be found using the cosine function:

cos(45) = adjacent / hypotenuse
cos(45) = x / 100

Solving for x, we get:
x = 100 * cos(45)
x ≈ 70.7107 centimeters

The vertical component of the first triangle can be found using the sine function:

sin(45) = opposite / hypotenuse
sin(45) = y / 100

Solving for y, we get:
y = 100 * sin(45)
y ≈ 70.7107 centimeters

Now let's move on to the second triangle formed by the path of 120 centimeters and the angle of reflection. Again, we know that the angle formed is 45 degrees. Using trigonometric functions, we can find the horizontal and vertical components of this triangle.

The horizontal component of the second triangle can be found using the cosine function:

cos(45) = adjacent / hypotenuse
cos(45) = x / 120

Solving for x, we get:
x = 120 * cos(45)
x ≈ 84.8528 centimeters

The vertical component of the second triangle can be found using the sine function:

sin(45) = opposite / hypotenuse
sin(45) = y / 120

Solving for y, we get:
y = 120 * sin(45)
y ≈ 84.8528 centimeters

Now, let's find the total horizontal and vertical distances traveled by the ball. The horizontal distance is the sum of the horizontal components from both triangles, which is 70.7107 + 84.8528 ≈ 155.5635 centimeters. The vertical distance is the sum of the vertical components from both triangles, which is 70.7107 + 84.8528 ≈ 155.5635 centimeters.

We can use the Pythagorean theorem to find the diagonal distance between the initial position of the ball and the corner pocket, which is the hypotenuse of a right triangle with the horizontal and vertical distances as its sides:

d^2 = (155.5635)^2 + (155.5635)^2
d^2 ≈ 24203.4381 + 24203.4381
d^2 ≈ 48406.8762
d ≈ √48406.8762
d ≈ 220 centimeters

Therefore, the initial position of the ball is approximately 220 centimeters away from the corner pocket.

To find the distance between the initial position of the ball and the corner pocket, we can use trigonometry.

First, let's break down the problem into two components: the distance traveled horizontally and the distance traveled vertically.

From the given information, we know that the ball traveled 100 centimeters before ricocheting off the rail and forming a 45-degree angle. We can calculate the horizontal distance traveled using the formula:

horizontal distance = distance traveled * cos(angle)

Using the given values:
horizontal distance = 100 centimeters * cos(45 degrees)

To calculate the cosine of 45 degrees, we can use a mathematical function or refer to a cosine table. The cosine of 45 degrees is √2 / 2.

horizontal distance ≈ 100 centimeters * (√2 / 2)
horizontal distance ≈ 100√2 / 2 centimeters

Simplifying further, we have:
horizontal distance ≈ 50√2 centimeters

Next, let's calculate the vertical distance traveled. We know that the ball traveled 120 centimeters into the corner pocket. Since the ball traveled vertically, we can use the formula:

vertical distance = distance traveled * sin(angle)

Using the given values:
vertical distance = 120 centimeters * sin(45 degrees)

The sine of 45 degrees is also √2 / 2.

vertical distance ≈ 120 centimeters * (√2 / 2)
vertical distance ≈ 120√2 / 2 centimeters
vertical distance ≈ 60√2 centimeters

Now, we can use the Pythagorean theorem to calculate the total distance between the initial position of the ball and the corner pocket. The theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides:

(total distance)^2 = (horizontal distance)^2 + (vertical distance)^2

Substituting the calculated values:
(total distance)^2 = (50√2)^2 + (60√2)^2
(total distance)^2 = 2500 + 3600
(total distance)^2 = 6100

Taking the square root of both sides:
total distance ≈ √6100
total distance ≈ 78.19 centimeters

Rounding to the nearest centimeter, the initial position of the ball is approximately 78 centimeters from the corner pocket where the ball was sunk.