Problem 12 - A Graphical Approach to College Algebra

Question

$$\begin{aligned} &\text {Solve each problem analytically, and support your solution}\\\ &\text {graphically.} \end{aligned}$$ World's Largest Easel The painting on the world's largest easel has a perimeter of 112 feet. Its length is 96 inches more than its width. What is its length in feet? (Source: Roadside America.)

Step-by-Step Solution

Answer

The length of the painting is 32 feet.

Step 1: Define the Variables

Let's define the variables for the dimensions of the painting. Let the width of the painting be \( w \) inches. According to the problem, the length is 96 inches more than the width, so the length can be expressed as \( l = w + 96 \) inches.

Step 2: Convert the Perimeter to Inches

The perimeter is given in feet, but since the dimensions are described in inches, convert the perimeter to inches. Since 1 foot equals 12 inches, the perimeter is \( 112 \times 12 = 1344 \) inches.

Step 3: Use the Perimeter Formula

The formula for the perimeter of a rectangle is \( 2(l + w) \). Substitute the expressions we have: \( 2(w + 96 + w) = 1344 \). Simplify the expression: \( 2(2w + 96) = 1344 \).

Step 4: Solve for the Width

Expand and solve the equation for \( w \):\[2(2w + 96) = 1344 \4w + 192 = 1344 \4w = 1152 \w = 288 \]

Step 5: Find the Length

Now that you have the width, you can find the length using the expression for the length.\[l = w + 96 = 288 + 96 = 384 \]

Step 6: Convert the Length to Feet

Since the calculations are in inches, convert the length back to feet for the final answer. \\( 384 \) inches divided by \( 12 \) equals \( 32 \) feet.

Key Concepts

Perimeter

The perimeter of a shape is the total distance around it. For rectangles, which have four sides, the perimeter is calculated by adding up the lengths of each side. Since opposite sides of a rectangle are equal, the perimeter formula simplifies to:

• For a rectangle: \( P = 2(l + w) \), where \( l \) is the length and \( w \) is the width.

In the context of the world's largest easel, the problem gives us the perimeter and asks us to find one of the dimensions. Understanding perimeter is crucial because it allows us to know the full boundary length of a rectangle when only some dimensions are known.
Therefore, getting the perimeter right is key to finding other missing dimensions like length and width.

Rectangle

A rectangle is a four-sided polygon, known as a quadrilateral, with opposite sides that are equal and parallel. It has two sets of equal lengths, making it both a versatile and frequently encountered shape in geometry.

• Characteristics include: four right angles, and the opposite sides are equal.

When dealing with problems involving rectangles, you'll often need to switch back and forth between the concepts of perimeter and area. Here, however, we are primarily interested in using the perimeter to determine the potential dimensions of the rectangle.
The equation given allows us to tie dimensions like width and length to calculate either one if the other is known or can be discovered.

Unit Conversion

Unit conversion involves changing a quantity measured in one unit into an equivalent quantity in another unit. For this exercise, converting feet to inches is crucial because the original problem gives the perimeter in feet, while the dimensions are given or calculated in inches. To make these conversions correctly:

• Remember: 1 foot equals 12 inches
• To convert feet to inches, multiply the number of feet by 12.
• To convert inches to feet, divide the number of inches by 12.

This conversion allows us to have the same units for easy and accurate calculations, as mismatched units can make error identification difficult.
Converting units is a common occurrence in algebra to maintain consistency and accuracy across equations.

Equation Solving

Equation solving is the process of finding the value of a variable that makes an equation true. It requires understanding the relationships expressed in the equation and performing algebraic manipulations to isolate the variable.

• To solve an equation: Rearrange the equation so that the variable appears on one side of the equation.
• Perform inverse operations to maintain the balance of the equation.

In this problem, equation solving is used to find the width of the rectangle first and then the length. By applying inverse operations, simple equations such as \( 2(2w + 96) = 1344 \) can be simplified step by step. Finally, solve for the variable by isolating it on one side of the equation.
This step-by-step method ensures accuracy and a clear path to finding the correct dimensions.