Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem analytically, and support your solution graphically. Perimeter of a Storage Shed \(\mathbf{A}\) carpenter must build a rectangular storage shed. She wants the length to be 3 feet greater than the width, and the perimeter must be 22 feet. Find the length and the width of the shed.

Step-by-Step Solution

Answer

Width is 4 feet, length is 7 feet.

Step 1: Define Variables

Let the width of the shed be denoted as \( w \). Since the length is 3 feet greater than the width, let the length be \( w + 3 \).

Step 2: Write the Perimeter Formula

The formula for the perimeter of a rectangle is given by: \[ P = 2 \times (\text{Length} + \text{Width}) \] Given that the perimeter \( P \) is 22 feet, substitute the known values into the formula: \[ 22 = 2 \times ((w + 3) + w) \].

Step 3: Simplify the Equation

Simplify the equation by combining like terms inside the parentheses: \[ 22 = 2 \times (2w + 3) \].

Step 4: Distribute the 2

Expand the equation by distributing 2: \[ 22 = 4w + 6 \].

Step 5: Solve for Width

Isolate \( w \) by subtracting 6 from both sides: \[ 16 = 4w \].Then divide both sides by 4 to solve for \( w \): \[ w = 4 \].

Step 6: Calculate the Length

Now that the width \( w \) is found, substitute it back into the expression for length: \( \text{Length} = w + 3 = 4 + 3 = 7 \).

Step 7: Confirm the Solution Graphically

Graphically verify by plotting a rectangle with width 4 and length 7, and check if the perimeter sums up to 22. The coordinates of the rectangle will be (0,0), (7,0), (7,4), and (0,4). Calculate the perimeter by adding up all sides: 7 + 4 + 7 + 4 = 22.

Key Concepts

Variables Definition

The first step in solving any mathematical problem is to define the variables. For this rectangle shed problem, we start by identifying what we do not know and assign variables accordingly.

• Width ( w ): The width of the shed, which is unknown at first. Therefore, we let this be our main variable.
• Length ( w + 3 ): Since the length is given as 3 feet greater than the width, the length is expressed as an additional 3 feet over the width variable.

Defining variables clearly and separately is crucial because it allows you to set the groundwork for inserting these into the equations you'll use later. It also keeps your work organized so that you don't make mistakes when plugging numbers into formulas.

Equation Simplification

Equation simplification helps transform the problem into something manageable. Following the perimeter formula for a rectangle, you start with the equation: \[ P = 2 \times (\text{Length} + \text{Width}) \]. For this shed problem, where the perimeter \( P \) is 22 feet, we substitute the values derived: \[ 22 = 2 \times ((w + 3) + w) \].

• Combine like terms:
The term \((w + 3 + w)\) within the brackets becomes \(2w + 3\).
• Distribution:
Multiply through by 2 to distribute across the equation: \(22 = 4w + 6\).
• Equation Resolution:
Solve for \(w\) by eliminating constants and isolating the variable: subtract the 6 on both sides to obtain \(16 = 4w\), then divide by 4 to get \(w = 4\).

Each simplification step converts a complex expression into an easier form, clearing the path progressively until the variable is isolated.

Graphical Verification

Once the solution is computed analytically, a graphical verification acts as a visual confirmation. Here, you check that your solution logically fits the criteria of the problem. By plotting a rectangle:

• Set one corner at the origin (0,0).
• Extend along the x-axis to (7,0) and along the y-axis to (0,4) based on calculated dimensions length = 7 and width = 4.
• The top-right corner will naturally sit at (7,4).

Verify by ensuring the perimeter calculation aligns as intended: add the lengths of all sides (7 + 4 + 7 + 4). The sum here should reaffirm the original perimeter of 22 feet. This graphical method serves as a reliable double-check since visualizing the dimensions can reveal errors that might have been made in earlier arithmetic steps.