Problem 101 - Algebra A Combined Function

Question

Recall that perimeter measures the distance around a plane figure and area measures the amount of surface of a plane figure. The expression \(2 l+2 w\) gives the perimeter of the rectangle below (measured in units), and the expression lw gives its area (measured in square units). Complete the chart below for the given lengths and widths. Be sure to include units. $$ \begin{array}{|l|c|c|c|} \hline \text { Length: } I & \text { Width: } \boldsymbol{w} & \begin{array}{c} \text { Perimeter of } \\ \text { Rectangle: } \\ \mathbf{2 I}+\mathbf{2 w} \end{array} & \begin{array}{c} \text { Area of } \\ \text { Rectangle: } \\ \boldsymbol{I} \boldsymbol{w} \end{array} \\ \hline 4 \text { in. } & 3 \text { in. } & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Perimeter: 14 inches; Area: 12 square inches.

Step 1: Identify Given Values

Identify the length \( l \) and width \( w \) of the rectangle. In this exercise, \( l = 4 \) inches and \( w = 3 \) inches.

Step 2: Calculate Perimeter

Use the formula for the perimeter of a rectangle, \( 2l + 2w \). Substitute the given values into the formula: \( 2(4) + 2(3) = 8 + 6 = 14 \) inches. Therefore, the perimeter is 14 inches.

Step 3: Calculate Area

Use the formula for the area of a rectangle, \( lw \). Substitute the given values into the formula: \( 4 imes 3 = 12 \) square inches. Therefore, the area is 12 square inches.

Key Concepts

Perimeter Calculation

To find the perimeter of a rectangle, it is essential to understand the concept of perimeter. The perimeter is the total length around the edges of a two-dimensional shape. For rectangles, calculating the perimeter is straightforward because it involves summing up the lengths of all four sides. Let's break it down step-by-step.

• Each rectangle has two pairs of opposite sides that are equal in length.
• To calculate the perimeter, you add up the lengths of these sides.
• The perimeter formula for a rectangle is expressed as \(2l + 2w\).

In this problem, if we are given a rectangle with a length \(l = 4\) inches and width \(w = 3\) inches, you plug these values into the formula:

\[2(4) + 2(3) = 8 + 6 = 14\]
The perimeter of the rectangle is 14 inches. Always remember to include units in your final answer.

Area Calculation

Understanding area is crucial when dealing with rectangles as it measures the surface encompassed within the perimeter. Calculating the area of a rectangle is straightforward once you get the hang of it.

• The area is the amount of space occupied by the rectangle.
• It is measured in square units because it represents two-dimensional space.

The formula for the area of a rectangle is simply length multiplied by width, or \(lw\). Given our rectangle with a length \(l = 4\) inches and width \(w = 3\) inches, we calculate the area as follows:

\[4 \times 3 = 12 \, \text{square inches}\]
Here, the area of the rectangle is 12 square inches. This means if you were to fill the rectangle with 1-inch by 1-inch tiles, you could fit 12 tiles in total. Again, don't forget the units in your answer.

Rectangle Properties

Rectangles are familiar shapes with unique properties that make them an interesting subject for geometric calculations. Understanding these properties aids in making logical assumptions and calculations surrounding rectangles.

• A rectangle is defined as a shape with four sides, where opposite sides are parallel and equal in length.
• Each internal angle in a rectangle is a right angle (90 degrees).
• Because both pairs of sides are equal, formulas related to perimeter and area remain consistent and are easy to remember.

The simple structure of a rectangle makes calculations predictable. For instance, knowing a rectangle's length and width enables you to determine all its geometric properties such as its perimeter, area, and sometimes even its diagonal using the Pythagorean theorem in more advanced calculus. Recognizing these properties not only helps in solving mathematical exercises but also in practical applications, like architectural planning and design.