Problem 24 - Contemporary Precalculus

Question

At Wrigley Field in Chicago, the straight-line distance from home plate over second base to the center field wall is 400 feet. How far is it from first base to the same point at the enter field wall? IHint; Adapt and extend the figure .

Step-by-Step Solution

Answer

Answer: The distance from first base to the center field wall at Wrigley Field is approximately 474 feet.

Step 1: Identifying the triangle sides

We will use the Pythagorean theorem to solve this problem in a right triangle. The three vertices of the triangle represent home plate (H), first base (F), and the center field wall (C). The hypotenuse is the distance HC = 400 feet, and we need to find the length of side FC. We also know that the side HF (distance from home plate to first base) is 90 feet.

Step 2: Finding the distance from home plate to second base

Since the baseball diamond is a square with side length 90 feet, we can use the Pythagorean theorem to find the distance from home plate to second base (HS). We have a right triangle with vertices at home plate (H), first base (F), and second base (S), with HF = FS = 90 feet. Using the Pythagorean theorem (a^2 + b^2 = c^2), we get: \(HS^2 = HF^2 + FS^2\) Plug in the values: \(HS^2 = 90^2 + 90^2\) Now, solve for HS: \(HS = \sqrt{90^2 + 90^2} = \sqrt{16200} = 90\sqrt{2}\) So, the distance from home plate to second base is \(90\sqrt{2}\) feet.

Step 3: Applying the Pythagorean theorem to find the distance from first base to the center field wall

Now, we've constructed a right triangle with vertices at first base (F), second base (S), and the center field wall (C), with FC being the unknown distance we need to find, FS = 90 feet, and SC = 400 feet (given). We can apply the Pythagorean theorem again (a^2 + b^2 = c^2), this time using FS and SC as our legs, and FC as our hypotenuse: \(FC^2 = FS^2 + SC^2\) Plug in the values: \(FC^2 = (90\sqrt{2})^2 + 400^2\) Now, solve for FC: \(FC = \sqrt{(90\sqrt{2})^2 + 400^2} = \sqrt{64800 + 160000} = \sqrt{224800}\) So, the distance from first base to the center field wall is \(\sqrt{224800}\) feet, approximately 474 feet.

Key Concepts

Understanding Right Triangles

In solving problems related to distances and certain figures, right triangles often play an essential role. A right triangle has one angle measuring exactly 90 degrees. This special characteristic makes it unique compared to other triangles because it allows us to use specific mathematical principles. One of the most fundamental theorems in mathematics, particularly when dealing with right triangles, is the Pythagorean theorem.

It states that for a right triangle with legs of lengths \(a\) and \(b\), and a hypotenuse of length \(c\), the sum of the squares of the legs is equal to the square of the hypotenuse:

• \(a^2 + b^2 = c^2\)

This relationship is always consistent, enabling us to find the length of any side if the other two sides are known. In our baseball field exercise, understanding the right triangle formed by home plate, first base, and center field wall helps us to find unknown distances using these principles.

Basics of Trigonometry

Trigonometry, a branch of mathematics, explores the relationships between the angles and sides of triangles, particularly right triangles. In our scenario, trigonometry isn't just about angles and triangles; it's also about dimensions and distances.

While the Pythagorean theorem is a foundation for understanding right triangles, trigonometry extends to include functions like sine, cosine, and tangent, which relate the ratios of a triangle’s sides to its angles. Although these functions aren't directly used in the solution of our problem, they illustrate how triangles' sides influence each other.

• Sine, cosine, and tangent can predict side lengths given angles.
• They offer deeper insight into solving various types of triangle problems.

In essence, a solid grasp of trigonometry broadens your problem-solving toolkit for questions involving measurements and spatial reasoning.

Geometry in Shapes and Figures

Geometry is the study of spaces and forms like points, lines, and shapes, such as polygons and circles. The baseball field problem is an excellent example of practical geometry. Our baseball field forms a diamond shape, which is essentially a square if we consider only the bases. The lines connecting these bases represent the edges of the square, each measuring 90 feet. To find distances related to this square, it’s crucial to understand geometric shapes’ properties.

• Each side of the baseball diamond is 90 feet, making it part of a perfect square.
• Geometry helps in visualizing the field as interconnected shapes -- squares and triangles.

Understanding these geometric relationships allows us to navigate through space effectively, be it in real-life sports fields or on paper, where these principles apply equally.

Practical Distance Calculation

The ability to calculate distance effectively is a crucial mathematical skill that extends beyond theoretical exercises into real-world applications. In our exercise involving the baseball field, distance calculation is not only essential for solving the problem but also invaluable for practical applications.Given the dimensions, such as the 90-foot sides of the baseball diamond and the 400-foot stretch from home plate over second base to center field, we use the Pythagorean theorem to calculate unknown distances. For instance:

• Calculate from home plate to second base: \(HS = 90\sqrt{2}\).
• Solving for the remaining distance from first base to center field wall utilizes these known values.

By breaking down these calculations, we leverage geometry and right triangles to determine precise measurements, making mathematical concepts applicable in fields such as engineering, physics, and everyday tasks like sports field layout management.