Problem 30 - Elementary and Intermediate Algebra

Question

The diagonal of a square parking lot is approximately \(1,414\) feet long. a. Find the length of one side of the parking lot. Round to the nearest foot. b. Find the approximate area of the parking lot.

Step-by-Step Solution

Answer

The side of the parking lot is approximately 1,000 feet. The area is approximately 1,000,000 square feet.

Step 1: Identify the formula for the diagonal of a square

The diagonal (\[d\]) of a square can be calculated using the formula \(d = s \sqrt{2}\), where \(s\) is the length of one side of the square.

Step 2: Solve for side length

Given that the diagonal is 1,414 feet, we can solve for \(s\) using the formula: \[1,414 = s \sqrt{2}\]Dividing both sides by \(\sqrt{2}\), we get: \[s = \frac{1,414}{\sqrt{2}}\]Calculating this gives \(s \approx 1,000\) feet.

Step 3: Calculate the area of the square

The area \(A\) of a square can be found using the formula \(A = s^2\). Already knowing that \(s \approx 1,000\), we calculate: \[A = (1,000)^2 = 1,000,000\] square feet.

Key Concepts

Diagonal of a Square

The diagonal of a square is a line segment that connects two opposite corners of the square. It divides the square into two equal right-angled triangles. To find the diagonal (\(d\)) of a square, you can use a simple formula: \[d = s \sqrt{2}\] Here, \(s\) is the length of one side of the square. This formula comes from applying the Pythagorean theorem to one of the right triangles, where the diagonal acts as the hypotenuse.

• The hypotenuse (diagonal) is always the longest side of the triangle.
• Knowing the diagonal allows you to determine the entire geometric proportions of the square easily.
• In a square, every diagonal is equal in length.

Area of a Square

The area of a square is an expression of how much space it covers on a flat surface. If you know the length of one side (\(s\)), you can calculate the area (\(A\)) through this straightforward formula: \[A = s^2\] This calculation is intuitive because a square is just a rectangle with all sides equal.

• The formula represents multiplying the side by itself, which gives the total number of square units inside the square.
• The area is always measured in square units (like square feet).
• This formula is fundamental for many practical applications, such as calculating the flooring for a room or the land area available for construction.

Knowing the area is crucial because it directly relates to how much space you have to use, share, or purchase. Calculating the area using square side measurements helps manage space efficiently.

Square Root Calculation

Calculating the square root is essential when dealing with geometry, particularly in finding values related to squares. The square root function, denoted as \(\sqrt{}\), is the inverse operation of squaring a number. For any given number \(x\), the square root provides a value \(y\) such that: \[y^2 = x\]

• In geometry, square roots often appear when solving for side lengths from diagonals, as seen in the diagonal formula \(s = \frac{d}{\sqrt{2}}\).
• This calculation helps simplify expressions and solve for unknown dimensions.
• Square roots are instrumental in trigonometry and algebra as well.

When tackling geometry problems, using the square root allows you to bridge the link between linear (side length) and area measurements, facilitating a comprehensive understanding of the geometric shape.