Question

A parking lot is \(144.3 \mathrm{~m}\) long and \(47.66 \mathrm{~m}\) wide. (a) What is the perimeter of the lot? (b) What is its area?

Short answer

(a) Perimeter is 383.92 m; (b) Area is 6875.538 m².

Step-by-step solution

Understanding Perimeter

The perimeter of a rectangle can be calculated using the formula: \[ P = 2(l + w) \] where \( l \) is the length and \( w \) is the width of the rectangle.

Calculating Perimeter

Substitute the given values into the formula for perimeter. Here, \( l = 144.3 \mathrm{~m} \) and \( w = 47.66 \mathrm{~m} \). So, \[ P = 2(144.3 + 47.66) \] Calculate the sum inside the parenthesis: \[ 144.3 + 47.66 = 191.96 \] Therefore, \[ P = 2 \times 191.96 = 383.92 \mathrm{~m} \] The perimeter of the lot is \( 383.92 \mathrm{~m} \).

Understanding Area

The area of a rectangle is calculated using the formula: \[ A = l \times w \] where \( l \) is the length and \( w \) is the width.

Calculating Area

Substitute the given values into the formula for area. Here, \( l = 144.3 \mathrm{~m} \) and \( w = 47.66 \mathrm{~m} \). So, \[ A = 144.3 \times 47.66 \] Perform the multiplication to find the area: \[ A = 6875.538 \mathrm{~m}^2 \] The area of the lot is \( 6875.538 \mathrm{~m}^2 \).

Key concepts

Perimeter calculation

Calculating the perimeter of a rectangle is a fundamental concept in geometry. It's crucial to understand it as it applies to many real-life situations such as determining the length of a fence needed to encircle a property. The formula for the perimeter of a rectangle is simple yet insightful:

• Formula: \( P = 2(l + w) \)
• Where \( P \) represents the perimeter, \( l \) is the length, and \( w \) is the width.

To calculate the perimeter, you add the length and the width of the rectangle together. This sum is then multiplied by two. Imagine if you walked around the edge of a parking lot, listing its measures, you'd cover the distance of all sides. This is what the formula simplifies: an effective way to know that entire distance quickly. For the parking lot example, we have a length of \( 144.3 \) meters and a width of \( 47.66 \) meters. Adding these gives \( 191.96 \) meters. Doubling this total provides the full perimeter: \( 383.92 \) meters. Using such calculations, you can easily plan around any rectangular area.

Area of rectangle

Understanding how to calculate the area of a rectangle is another key skill in geometry. It helps you know precisely how much space within a boundary is available, which is helpful for planning layout designs, like where to park cars in a parking lot. The technique involves a straightforward mathematical formula:

• Formula: \( A = l \times w \)
• Where \( A \) is the area, \( l \) is the length, and \( w \) is the width.

To visualize, think of filling the rectangle with square tiles, whereby each tile's side measures one meter. The total number of tiles used gives you the area in square meters. This formula effectively multiplies the length and the width, representing those tiles.For the exercise here, the parking lot's length and width were \( 144.3 \) meters and \( 47.66 \) meters respectively. By multiplying these two numbers, you compute an area of \( 6875.538 \) square meters, showing exactly how much flat space the lot occupies. Such insights assist in maximizing the space usage efficiently.

Mathematics problem-solving

Tackling mathematical problems, especially in geometry, is not just about plugging numbers into formulas. It’s about thinking logically and applying concepts to reach accurate conclusions. Here's a structured approach beneficial when working on shape-related exercises:

• Identify what is being asked. Clearly understand the problem’s requirements before proceeding.
• Equip yourself with the right formulas. Know which mathematical equations relate to the dimensions you're analyzing.
• Substitute the correct values and ensure calculations are accurate.

Problem-solving in mathematics involves progressing from understanding the concept to applying that understanding practically with numbers. This journey sharpens your analytical skills and bolsters your ability to handle related tasks in everyday scenarios. Hence, in problems regarding perimeters and areas, the formulas used aren't just memorization tasks; they are tools that link mathematical theory to pragmatic use cases. Approach each problem with a strategic mindset: start by dissecting it into simpler parts, then process those parts systematically. This helps in yielding precise and useful results, just like solving for the perimeter and area in the examples above.