Problem 104 - 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. $$ \begin{array}{|l|c|c|c|} \hline \text { Length: } I & \text { Width: } \boldsymbol{w} & \begin{array}{c} \text { Perimeter of } \\ \text { Rectangle: } \\ \mathbf{2 I}+\mathbf{2 w} \end{array} & \begin{array}{c} \text { Area of } \\ \text { Rectangle: } \\ \boldsymbol{I} \boldsymbol{w} \end{array} \\ \hline 4.6 \text { in. } & 2.4 \text { in. } & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Perimeter: 14 in; Area: 11.04 in².

Step 1: Identify Variables

We need to identify the given values for length \( l \) and width \( w \). From the table, the length \( l = 4.6 \) inches and width \( w = 2.4 \) inches.

Step 2: Calculate Perimeter

Use the formula for the perimeter of a rectangle: \( 2l + 2w \). Substitute the given values: \( 2(4.6) + 2(2.4) \). This simplifies to \( 9.2 + 4.8 = 14 \) inches.

Step 3: Calculate Area

Use the formula for the area of a rectangle: \( lw \). Substitute the given values: \( 4.6 \times 2.4 \). This simplifies to \( 11.04 \) square inches.

Key Concepts

Geometry

Geometry is a branch of mathematics that focuses on the properties and relations of points, lines, surfaces, and solids. It's like the blueprint of the mathematical world, providing a framework to understand the shape and size of everything around us. When we talk about geometry in relation to plane figures like rectangles, we're concerned with their areas and perimeters.
For instance, a rectangle is a two-dimensional shape with opposite sides that are equal in length. This simple yet versatile shape is everywhere in our daily lives; from smartphone screens to sheets of paper. Understanding how to calculate both the perimeter and area of a rectangle is essential for solving practical problems, such as determining how much material is needed for a frame or how much space a tile will cover.

Rectangle

A rectangle is a four-sided polygon, which is also known as a quadrilateral. Each angle of a rectangle is a right angle, which measures 90 degrees. A distinctive feature of rectangles is that they have opposite sides that are parallel and equal in length. This makes working with rectangles particularly straightforward when performing calculations.
The perimeter of a rectangle is the total distance around its edges. Since the rectangle has two pairs of equal sides, you can find the perimeter by adding up all the sides or by using the formula \[ 2l + 2w \] where \( l \) is the length and \( w \) is the width. On the other hand, the area of a rectangle is the measure of the space taken up by the rectangle, which can be found using the formula \[ lw \]. This calculation is essential for things like determining how many square tiles you need to cover a floor.

Mathematical Formulas

Mathematical formulas are the language of mathematics that help us make calculations easier and more efficient. These formulas are like recipes that provide us with a step-by-step process to solve a problem. When it comes to rectangles, two crucial formulas come into play: one for the perimeter and one for the area.

• The formula for the perimeter of a rectangle is \[ 2l + 2w \]. This formula helps in determining how much material you need to go around the rectangle.
• The formula for the area of a rectangle is \[ lw \]. This gives us the measure of the space inside the rectangle, which is crucial for tasks such as carpeting a room.

Both these formulas are straightforward but require careful substitution of given values to ensure accuracy. By following these formulas, you can efficiently calculate the dimensions needed for any rectangular space or object.