The four bases of a baseball diamond form a square 90 feet on a side. The shortstop,S, is in a position that is 50 feet from second bases and forms a 15 degree angle with the base path as shown. Find the distance between the shortstop and first base to the nearest foot
To find the distance between the shortstop and first base, we can use trigonometry.
First, let's label the points on the diagram. We have:
- Shortstop (S)
- Second base (2B)
- First base (1B)
We can create a right triangle using the shortstop (S), second base (2B), and first base (1B). The hypotenuse of this triangle will be the distance between the shortstop and first base.
To find the length of the adjacent side of the triangle (the distance between the shortstop and second base), we know that it forms a 15-degree angle with the base path, and its length is given as 50 feet.
Using trigonometry, the adjacent side can be found using the formula:
adjacent = hypotenuse * cos(angle)
In this case, the adjacent side is the distance between the shortstop and second base:
adjacent = 50 * cos(15°)
We can now calculate the adjacent side:
adjacent = 50 * cos(15°) ≈ 47.87 feet (approx.)
Next, to find the length of the hypotenuse (the distance between the shortstop and first base), we need to use the Pythagorean theorem:
hypotenuse^2 = adjacent^2 + opposite^2
Since the distance between the shortstop and first base is the hypotenuse, we will be solving for it.
Plugging in the values we have:
hypotenuse^2 = (47.87)^2 + (90)^2
Simplifying this equation:
hypotenuse^2 = 2285.49 + 8100
hypotenuse^2 = 10385.49
Taking the square root of both sides to find the hypotenuse:
hypotenuse ≈ √10385.49 ≈ 101.9 feet (approx.)
Therefore, the distance between the shortstop and first base is approximately 101.9 feet.