Problem 102 - 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. $$ \begin{array}{|l|c|c|c|} \hline \text { Length: } I & \text { Width: } \boldsymbol{w} & \begin{array}{c} \text { Perimeter of } \\ \text { Rectangle: } \\ \mathbf{2 I}+\mathbf{2 w} \end{array} & \begin{array}{c} \text { Area of } \\ \text { Rectangle: } \\ \boldsymbol{I} \boldsymbol{w} \end{array} \\ \hline 6 \text { in. } & 1 \text { in. } & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Perimeter: 14 inches, Area: 6 square inches.

Step 1: Understanding Perimeter

The perimeter of a rectangle is found using the formula \( P = 2l + 2w \), where \( l \) is the length and \( w \) is the width.

Step 2: Understanding Area

The area of a rectangle is calculated using the formula \( A = lw \), where \( l \) is the length and \( w \) is the width.

Step 3: Substitute Length and Width for Perimeter

Substitute the given values into the perimeter formula. With \( l = 6 \) inches and \( w = 1 \) inch, the perimeter calculation becomes \( P = 2(6) + 2(1) \).

Step 4: Calculate Perimeter

Perform the calculation: \( P = 12 + 2 = 14 \) inches. This is the total perimeter of the rectangle.

Step 5: Substitute Length and Width for Area

Substitute the same values into the area formula. With \( l = 6 \) inches and \( w = 1 \) inch, the area calculation becomes \( A = 6 \times 1 \).

Step 6: Calculate Area

Perform the calculation: \( A = 6 \) square inches. This is the total area of the rectangle.

Step 7: Fill in the Chart

Using the calculations above, fill in the chart: Perimeter is 14 inches and Area is 6 square inches.

Key Concepts

Understanding Geometry

Geometry is a branch of mathematics that deals with the properties and relations of points, lines, surfaces, and solids. It helps in understanding the shape and size of objects, which is essential when calculating figures like area and perimeter. In the context of rectangles, geometry teaches us how two-dimensional shapes like rectangles are measured and analyzed. This can be quite intuitive because rectangles are defined by just their length and width.

• A rectangle is a four-sided shape where every angle is a right angle (90 degrees).
• The opposite sides of a rectangle are equal in length.
• Geometry helps us to visualize and calculate real-world spaces, vital for tasks like construction, design, and art.

The simple nature of rectangles makes geometric calculations straightforward, allowing students to focus on understanding basic concepts without complicated variables.

Properties and Characteristics of Rectangles

Rectangles are some of the most common shapes studied in geometry due to their straightforward properties. Understanding these properties is critical for calculating both the perimeter and the area.

• The length ( l ) is the longer side of the rectangle.
• The width ( w ) is the shorter side of the rectangle.
• Rectangles have parallel sides, meaning each pair of opposite sides is equal.

These properties mean that all rectangles have certain rules that govern their perimeters and areas. Recognizing and applying these properties allows students to quickly solve geometric problems related to rectangles and build a strong foundation for more advanced geometry topics.

Mathematics Formulas for Perimeter and Area

Mathematics provides us with formulas to easily calculate the perimeter and area of rectangles, essential information in fields like architecture, engineering, and graphic design. To find the perimeter of a rectangle, use the formula:\[P = 2l + 2w\]This formula adds up all sides, illustrating the complete distance around the rectangle. For the area, the calculation is done by:\[A = lw\]This formula multiplies the rectangle's length by its width, showing the total square unit area within the rectangle.

• Example: With a length of 6 inches and a width of 1 inch, the perimeter becomes \( P = 2(6) + 2(1) = 14 \) inches.
• The area becomes \( A = 6 \times 1 = 6 \) square inches.

These formulas are quick to use and provide clear, direct calculations that are important for solving practical problems. Familiarity with these equations makes working with rectangles an easy and systematic process, paving the way to more complex geometric calculations.