Problem 50 - Algebra and Trigonometry

Question

According to Little League baseball official regulations, the diamond is a square 60 feet on a side. The pitching rubber is located 46 feet from home plate on a line joining home plate and second base (a) How far is it from the pitching rubber to first base? (b) How far is it from the pitching rubber to second base? (c) If a pitcher faces home plate, through what angle does he need to turn to face first base?

Step-by-Step Solution

Answer

a) ~75.6 feet; b) 14 feet; c) ~52.2°

Step 1 - Understanding the Diamond

The baseball diamond is a square with each side measuring 60 feet. Home plate, first base, second base, and third base are located at the vertices of the square.

Step 2 - Locating the Pitching Rubber

The pitching rubber is 46 feet from home plate along the diagonal line from home plate to second base.

Step 3 - Distance from Pitching Rubber to First Base

Using the Pythagorean theorem, find the distance from the pitching rubber to first base. The pitching rubber forms a right triangle with home plate and first base where the legs are the distances from home plate to the pitching rubber (46 feet) and from home plate to first base (60 feet).

Step 4 - Applying the Pythagorean Theorem

Apply the Pythagorean theorem: \[ c = \sqrt{a^2 + b^2} \] where \(a = 46\) feet and \(b = 60\) feet. So, \[ c = \sqrt{46^2 + 60^2} = \sqrt{2116 + 3600} = \sqrt{5716} \approx 75.6 \text{ feet} \]

Step 5 - Calculate the Distance from Pitching Rubber to Second Base

Since the pitching rubber is on the line between home plate and second base, the distance from the pitching rubber to second base is simply: \[ 60 - 46 = 14 \text{ feet} \]

Step 6 - Angle to First Base

Use trigonometry to determine the angle. The pitching rubber, home plate, and first base form a right triangle. The angle at the pitching rubber facing home plate is \( \theta \). We use the tangent function: \[ \tan^{-1} \left( \frac{60} {46} \right) \] Calculate: \[ \theta = \tan^{-1} \left( \frac{60}{46} \right) \approx 52.2° \]

Key Concepts

Pythagorean Theorem

To solve the given problem, understanding the Pythagorean theorem is essential. The Pythagorean theorem is used in right-angled triangles to relate the lengths of the sides. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written mathematically as: \[ c^2 = a^2 + b^2 \] Here, 'c' is the hypotenuse, and 'a' and 'b' are the other two sides. In this problem, we use this theorem to find the distance from the pitching rubber to first base by treating it as a right triangle's hypotenuse.

Distance Calculation

Distance calculation is crucial in solving problems involving geometry and real-world applications. In the given problem, we need to calculate the distances between different points on a baseball diamond. To find the distance from the pitching rubber to first base, we use the Pythagorean theorem. Here, 'a' is the distance from home plate to the pitching rubber (46 feet), and 'b' is the distance from home plate to first base (60 feet). By substituting these values into the Pythagorean theorem, we get: \[ c = \sqrt{46^2 + 60^2} = \sqrt{2116 + 3600} = \sqrt{5716} \approx 75.6 \text{ feet} \] For the distance from the pitching rubber to second base, we simply subtract the distance from the pitching rubber to home plate (46 feet) from the total distance between home plate and second base (60 feet): \[ 60 - 46 = 14 \text{ feet} \]

Trigonometric Functions

Trigonometric functions help us find angles and distances in right triangles. In this problem, we use the tangent function to calculate the angle at the pitching rubber when a pitcher faces home plate and turns to face first base. The tangent function relates the opposite and adjacent sides of a right triangle. For angle \( \theta \) at the pitching rubber, the opposite side is the distance from home plate to first base (60 feet), and the adjacent side is from home plate to the pitching rubber (46 feet). We use: \[ \theta = \tan^{-1} \left( \frac{60}{46} \right) \approx 52.2° \] This tells us that the pitcher needs to turn approximately 52.2 degrees to face first base from home plate.

Geometry in Real-world Applications

Geometry is widely used in real-world applications like sports, construction, and design. In this problem, understanding the geometry of a baseball diamond is crucial. A baseball diamond is a square with each side measuring 60 feet. By using geometric principles, we can solve for distances and angles within the diamond. In this particular case, the baseball diamond forms a real-world scenario where the Pythagorean theorem and trigonometric functions are applied to calculate distances from specific points like the pitching rubber to bases and the angles of turns. This application of geometry helps understand how mathematical concepts are used in sports to determine optimal plays and positions.

Right Triangle Properties

Understanding the properties of right triangles is key to solving geometrical problems. A right triangle has one angle equal to 90 degrees, and several important properties stem from this. In the context of the baseball problem, the right triangle involves the pitching rubber, home plate, and first base. Here, the legs of the triangle are the distances from the home plate to the pitching rubber (46 feet) and from home plate to first base (60 feet). The hypotenuse is the distance from the pitching rubber to first base. Using these properties, combined with the Pythagorean theorem and trigonometric functions, allows for the calculations and understanding of geometric principles in a structured way. The calculation of the angle using the tangent function also relies on the properties of right triangles.