Problem 30 - Algebra and Trigonometry

Question

Dimensions of a Lot A half-acre building lot is five times as long as it is wide. What are its dimensions? [Note: 1 acre \(=43,560 \mathrm{ft}^{2} . ]\)

Step-by-Step Solution

Answer

The lot is approximately 66 feet wide and 330 feet long.

Step 1: Understand the Problem

We need to find the dimensions of a half-acre lot, which is five times as long as it is wide. We know that 1 acre equals 43,560 square feet, so half an acre is 21,780 square feet.

Step 2: Set Up an Equation

Let the width of the lot be \( w \) feet. Since the length is five times the width, the length is \( 5w \) feet. The area of the lot is given by its length times its width, which is \( 5w \times w = 5w^2 \) square feet. We know that this area equals 21,780 square feet.

Step 3: Solve the Equation for Width

We have the equation \( 5w^2 = 21,780 \). Dividing both sides by 5 gives \( w^2 = 4,356 \). Taking the square root of both sides, we find \( w = \sqrt{4,356} \approx 66 \) feet.

Step 4: Calculate the Length

Since the length is five times the width, the length of the lot is \( 5 \times 66 = 330 \) feet.

Key Concepts

Understanding Area Calculation

Area calculation is a fundamental concept in mathematics, especially when dealing with real-world problems like measuring land. The area of a shape quantifies the space it covers. Often measured in square units, calculating area requires specific formulas based on the shape or figure in question.

In this context, we're calculating the area of a rectangular lot. The formula to find the area of a rectangle is given by multiplying its length by its width:

• Area = Length × Width

In the exercise, we have a lot which covers half an acre, equivalent to 21,780 square feet, showcasing how area units can be converted and applied. Understanding the conversion is crucial for accurate calculations in various measurement units.

Solving Equations for Unknown Dimensions

Equation solving is an essential skill in algebra that allows us to find unknown values using known ones. This involves performing operations to isolate the unknown variable on one side of an equation.

In the problem at hand, the unknown variable is the lot's width, represented by \( w \). We know:

• The total area of the lot equals 21,780 square feet.
• The relationship between width \( w \) and length, with length being five times the width, expressed as \( 5w \).

By setting up the equation for the area, \( 5w^2 = 21,780 \), we solved for \( w \) by performing the following steps:

• Divide both sides by 5: \( w^2 = 4,356 \).
• Take the square root of both sides: \( w = \sqrt{4,356} \approx 66 \) feet.

Equation solving helped us determine the dimensions of the lot starting from the area and using its length-to-width ratio.

Basic Concepts of Geometry

Geometry deals with shapes, sizes, and the properties of space. Understanding geometric concepts is fundamental for issues related to design, construction, and even navigation. Basic geometric formulas help solve everyday problems efficiently.

The exercise involves a rectangular piece of land, showcasing two crucial geometric aspects:

• Shape Identification: Identifying the shape as a rectangle and applying corresponding formulas.
• Proportional Relationships: Knowing that the length is a multiple of the width, specifically five times, illustrates how understanding properties of shapes simplifies problem-solving.

Geometry not only aids in solving complex mathematical problems but also provides a framework that applies to countless real-world situations, like designing plots of land to specific dimensions.