Problem 103 - 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. $$ \begin{array}{|l|c|c|c|} \hline \text { Length: } I & \text { Width: } \boldsymbol{w} & \begin{array}{c} \text { Perimeter of } \\ \text { Rectangle: } \\ \mathbf{2 I}+\mathbf{2 w} \end{array} & \begin{array}{c} \text { Area of } \\ \text { Rectangle: } \\ \boldsymbol{I} \boldsymbol{w} \end{array} \\ \hline 5.3 \text { in. } & 1.7 \text { in. } & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Perimeter: 14 in., Area: 9.01 sq. in.

Step 1: Understand the Formulas

The perimeter of a rectangle is calculated with the formula \( 2l + 2w \), where \( l \) is the length and \( w \) is the width. The area of a rectangle is determined with the formula \( lw \). We'll use these formulas to fill in the values for the given dimensions.

Step 2: Substitute Values into Perimeter Formula

Using the formula \( 2l + 2w \) for the perimeter, substitute \( l = 5.3 \) inches and \( w = 1.7 \) inches. Therefore, compute \( 2 \times 5.3 + 2 \times 1.7 \).

Step 3: Calculate Perimeter

Calculate \( 2 \times 5.3 = 10.6 \) inches and \( 2 \times 1.7 = 3.4 \) inches. Add these two results to obtain the perimeter: \( 10.6 + 3.4 = 14 \) inches.

Step 4: Substitute Values into Area Formula

Using the formula \( lw \) for the area, substitute \( l = 5.3 \) inches and \( w = 1.7 \) inches. Therefore, compute \( 5.3 \times 1.7 \).

Step 5: Calculate Area

Calculate \( 5.3 \times 1.7 = 9.01 \) square inches to find the area.

Step 6: Complete the Chart

Insert the calculated perimeter "14 in." and area "9.01 sq. in." into the chart to complete it.

Key Concepts

Understanding the Perimeter Formula

The perimeter of a rectangle is a fundamental mathematical concept that reflects the total distance around its edge. For rectangles, this calculation is made simpler with the perimeter formula:

• Perimeter formula for rectangles: \( P = 2l + 2w \)

Here, \( l \) stands for the length, and \( w \) denotes the width. By multiplying these dimensions by two and adding them together, we can easily find the perimeter. For example, if the length is 5.3 inches and the width is 1.7 inches:

• First, calculate \( 2 \times 5.3 = 10.6 \) inches
• Next, calculate \( 2 \times 1.7 = 3.4 \) inches
• Finally, add these values: \( 10.6 + 3.4 = 14 \) inches

Thus, the perimeter of this rectangle would be 14 inches. This formula provides a quick and straightforward way to determine the perimeter, making it very useful for various practical applications.

Exploring the Area Formula

Understanding how to calculate the area of a rectangle is another vital skill in mathematics. The area formula for rectangles calculates the size of the surface enclosed within the edges:

• Area formula for rectangles: \( A = lw \)

In this formula, \( l \) and \( w \) again represent the length and width, respectively. By multiplying these two dimensions, we can find the total area. For instance, given a length of 5.3 inches and a width of 1.7 inches:

• Calculate: \( 5.3 \times 1.7 = 9.01 \) square inches

The result, 9.01 square inches, represents the area of the rectangle. This formula is crucial for tasks where the measurement of space is needed, such as in interior design, architecture, and more. It allows for the quick determination of how much space an object or surface covers.

Mathematical Problem Solving with Rectangles

Solving mathematical problems involving rectangles often require a robust understanding of the perimeter and area formulas. To effectively tackle such problems:

• First, ensure the correct dimensions are known: identify the length (\( l \)) and width (\( w \)) of the rectangle.
• Apply the respective formulas to calculate the perimeter and area: use \( P = 2l + 2w \) for perimeter and \( A = lw \) for area.
• Substitute the given dimensions into the formulas accurately, performing multiplication and addition as needed.

For example, with a rectangle of length 5.3 inches and width 1.7 inches, the process would be:

• Calculate the perimeter: \( 2 \times 5.3 + 2 \times 1.7 = 14 \) inches
• Calculate the area: \( 5.3 \times 1.7 = 9.01 \) square inches

This step-by-step approach provides a structured method that can be applied to a wide range of rectangle-related problems, enhancing problem-solving efficiency and accuracy in mathematics.