Problem 141 - College Algebra

Question

Use the Pythagorean Theorem and the square root property to solve Exercises \(140-143 .\) Express answers in simplificd radical form. Then find a decimal approximation to the nearest tenth. A rectangular park is 4 miles long and 2 miles wide. How long is a pedestrian route that runs diagonally across the park?

Step-by-Step Solution

Answer

The pedestrian route that runs diagonally across the park is \(2\sqrt {5}\) miles, approximately 4.5 miles.

Step 1: Identify the dimensions

Identify the sides of the rectangular park given in the problem. The length is 4 miles and the width is 2 miles.

Step 2: Apply the Pythagorean theorem

Apply the Pythagorean theorem \(c^{2} = a^{2} + b^{2}\) where a = 4 and b = 2. Therefore \(c^{2} = (4)^{2} + (2)^{2}.\)

Step 3: Calculate the diagonal's square

Calculate the sum on the right side of the equation. \(c^{2} = 16 + 4 = 20.\)

Step 4: Find the diagonal length

Take the square root of both sides to obtain the length of the diagonal. So, \(c = \sqrt{20}\)

Step 5: Simplify the radical

Simplify \( \sqrt{20}\) to its simplest radical form which is \(2\sqrt {5}\).

Step 6: Approximate to a decimal

Find a decimal approximation of \(c = 2\sqrt {5}\) to the nearest tenth, which is 4.5 miles.

Key Concepts

Understanding the Square Root Property

The square root property is a key to unlocking the length of components within geometric shapes, such as the diagonal in a rectangle. This property states that if you have an equation where a variable is squared, like \(x^2 = y\), the solution can be found by applying the square root to both sides of the equation. This yields \(x = \pm \sqrt{y}\). For practical purposes, we often consider only the positive square root when dealing with lengths in geometry.

In the context of the exercise, once we have calculated the square \(c^2 = 20\) for the diagonal of the rectangle, applying the square root property helps us find \(c = \sqrt{20}\). This solves the puzzle of determining the actual length of the diagonal.

Simplified Radical Form

Simplifying a radical means finding the simplest form of a square root that still holds the same value. It involves breaking down the number inside the square root into its prime factors. This makes it easier to manage and understand.

For \(\sqrt{20}\), the breakdown is as follows: \(20 = 2 \times 2 \times 5\). Here, the pair of 2s can be taken outside of the square root. It results in \(2\sqrt{5}\), which is the simplified radical form. This not only makes calculations more straightforward but also conforms to the required answer format in many math problems.

Finding Decimal Approximation

A decimal approximation provides a more tangible understanding of a radical form by converting it into a number one is likely more familiar with. When you simplify a radical and need to express the value in decimals, rounding becomes essential to present the number to a certain precision.

In our exercise, to find the diagonal's length in decimal form, we approximate \(2\sqrt{5}\). This equals approximately 4.472, rounded to the nearest tenth, it is 4.5. This approach helps especially in practical scenarios, such as figuring out approximate distances in real life, where decimals are often more useful than radical expressions.

Diagonal of a Rectangle

Understanding the concept of a diagonal is crucial in geometry. For a rectangle, the diagonal is a line segment that connects opposite corners, effectively splitting the rectangle into two congruent right triangles.

Using the Pythagorean Theorem, which relates the side lengths of a right triangle, we determine the diagonal's length via \(c^2 = a^2 + b^2\). Here, \(a\) and \(b\) are the length and width of the rectangle, enabling us to solve \(c\), the diagonal.

• For example, in our park problem, with sides of 4 miles and 2 miles, the diagonal \(c\) is found using \(c = \sqrt{4^2 + 2^2}\).
• This calculation yields a diagonal \(c = \sqrt{20}\), showing how diagonals span across spaces, providing minimum distance routes.

This principle is foundational in understanding not only mathematical theory but also practical applications like navigating the shortest pathways.