103. A billiard ball, aimed at a point A on the edge of the cushion of a horizontal billiard table, rolls along a line making 18 degrees with the cushion. If the cushion over hangs so that its edge is 4 cm. above the table and the diameter of the ball is 5 cm. find the distance (measured along the edge of the cushion) between the point A and the point where the ball strikes the cushion.

Question

Visualize a horizontal billiard table from a slightly elevated angle. Display a billiard ball on its surface, aligned such that its trajectory forms an 18 degrees angle with one of the table's cushions. Make sure that the cushion clearly overhangs the table by 4 cm. Depict the diameter of the billiard ball as 5 cm. Illustrate a point A on the edge of the cushion where the billiard ball is aimed and allow a visible, measured distance along the edge of the cushion between point A and the expected point of contact of the ball with the cushion. No text allowed.

103. A billiard ball, aimed at a point A on the edge of the cushion of a horizontal billiard table, rolls along a line making 18 degrees with the cushion. If the cushion over hangs so that its edge is 4 cm. above the table and the diameter of the ball is 5 cm. find the distance (measured along the edge of the cushion) between the point A and the point where the ball strikes the cushion.

Answer 1

To find the distance between point A and the point where the ball strikes the cushion, we need to use the information given and apply some trigonometry.

First, let's visualize the situation.

We have a billiard ball rolling towards point A on the edge of the cushion. The cushion overhangs the table, and we need to find the distance between point A and the point where the ball strikes the cushion.

Let's break down the problem into steps:

Step 1: Calculate the height of the cushion above the table:
Since the cushion overhangs the table by 4 cm, the height of the cushion above the table is 4 cm.

Step 2: Calculate the distance between the point A and the point where the ball strikes the cushion:
From the given information, we know that the line along which the ball is rolling makes an angle of 18 degrees with the cushion.

To calculate the distance along the cushion, we can use trigonometry. In this case, we'll use the tangent function.

Tangent is defined as the ratio of the opposite side (height of the cushion) to the adjacent side (distance along the cushion).

So, we have:
tan(18 degrees) = height of cushion / distance along the cushion

Rearranging the equation, we can solve for the distance along the cushion:
distance along the cushion = height of cushion / tan(18 degrees)

Substituting the values:
distance along the cushion = 4 cm / tan(18 degrees)

Step 3: Calculate the answer:
Now, we can use a scientific calculator to find the value of tan(18 degrees) and substitute it back into our equation.

After evaluating, we find that the value of tan(18 degrees) is approximately 0.3249.

Substituting this value back into the equation, we get:
distance along the cushion = 4 cm / 0.3249

Evaluating the expression, we find that the distance along the cushion is approximately 12.30 cm.

Therefore, the distance between point A and the point where the ball strikes the cushion is approximately 12.30 cm (measured along the edge of the cushion).