Problem 3 - College Algebra with Modeling and Visualization

Question

Find the area and perimeter of the rectangle with length \(L\) and width \(W\). \(L=100\) meters, \(W=35\) meters

Step-by-Step Solution

Answer

The area is 3500 square meters, and the perimeter is 270 meters.

Step 1: Understand the Formula for Perimeter

The perimeter of a rectangle is the total distance around the outer edge. It can be calculated using the formula: \( P = 2L + 2W \), where \(L\) is the length and \(W\) is the width.

Step 2: Calculate the Perimeter

Using the given values \(L = 100\) meters and \(W = 35\) meters, substitute these into the formula: \[ P = 2(100) + 2(35) = 200 + 70 = 270 \]So, the perimeter of the rectangle is 270 meters.

Step 3: Understand the Formula for Area

The area of a rectangle is the amount of space inside it. It can be calculated using the formula: \( A = L \times W \), where \(L\) is the length and \(W\) is the width.

Step 4: Calculate the Area

Using the given values \(L = 100\) meters and \(W = 35\) meters, substitute these into the formula: \[ A = 100 \times 35 = 3500 \]So, the area of the rectangle is 3500 square meters.

Key Concepts

Area Calculation

Understanding how to calculate the area of a rectangle is essential as it's the measure of the space inside the rectangle. To determine the area, you need to use the formula:

• Formula: The area (\( A \)) is calculated as the product of the length (\( L \)) and the width (\( W \)).

So, the formula is \( A = L \times W \).
For instance, if the length is 100 meters and the width is 35 meters, then the area is calculated as follows:

• \( A = 100 \times 35 = 3500 \)

Thus, the area of the rectangle is 3500 square meters. This means the space enclosed within the rectangle covers 3500 square meters. Calculating the area can help in real-life scenarios, such as determining the amount of carpet required to cover a floor. Simply put: measure, multiply, and find the area!

Perimeter Calculation

When it comes to perimeter calculation, think of it as finding the total edge length of the rectangle. The perimeter is essentially the total distance around the rectangle and it requires the sum of all sides.

• Formula: The perimeter (\( P \)) is given by \( P = 2L + 2W \), where \( L \) is the length and \( W \) is the width.

Using this knowledge, consider a rectangle with a length of 100 meters and a width of 35 meters:

• Substitute into the formula: \( P = 2(100) + 2(35) = 200 + 70 = 270 \)

Consequently, the perimeter amounts to 270 meters. Understanding perimeter is practically useful, for example, when you need to enclose a garden with fencing. Remember, perimeter is all about the boundary; add up all sides to get it accurately.

Rectangles

Rectangles are four-sided shapes or quadrilaterals characterized by their opposite sides being equal and parallel. The distinct feature of rectangles is that they have right angles at each corner. Therefore, every angle in a rectangle is 90 degrees.

• Properties: Rectangles have two pairs of equal sides, with one pair being the length (\( L \)) and the other the width (\( W \)).
• Each internal angle is a right angle (90 degrees).
• Opposite sides are parallel, which is why the calculations for perimeter and area are straightforward and reliable.

In practical scenarios, rectangles are everywhere: books, screens, tables, and more. This geometry is not only fundamental in academic pursuits but also deeply integrated into our daily life objects. By understanding rectangles, you're better equipped to solve real-world problems and geometry-related questions more easily and accurately.