Problem 10 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem analytically, and support your solution graphically. Dimensions of a Label The length of a rectangular mailing label is 3 centimeters less than twice the width. The perimeter is 54 centimeters. Find its dimensions.

Step-by-Step Solution

Answer

The dimensions are: width = 10 cm, length = 17 cm.

Step 1: Define Variables

Let the width of the label be denoted by \( w \). According to the problem, the length \( l \) of the label is 3 cm less than twice the width. Therefore, we can express the length as \( l = 2w - 3 \).

Step 2: Create the Perimeter Equation

The perimeter of a rectangle can be calculated using the formula \( P = 2(l + w) \). Given that the perimeter is 54 cm, substitute the expressions for \( l \) and \( w \) into the equation: \( 54 = 2((2w - 3) + w) \).

Step 3: Simplify and Solve for Width

Simplify the equation: \( 54 = 2(3w - 3) \). Distributing the 2 gives \( 54 = 6w - 6 \). Add 6 to both sides to obtain \( 60 = 6w \). Divide both sides by 6 to find \( w = 10 \) cm.

Step 4: Calculate the Length

Using the expression for \( l \) from Step 1, substitute \( w = 10 \) cm to calculate the length: \( l = 2(10) - 3 = 20 - 3 = 17 \) cm.

Step 5: Solution Verification Graphically

On a graph, plot the equations \( P = 2((2w - 3) + w) = 54 \) and visually check if the intersection point aligns with \( w = 10 \), \( l = 17 \). This confirms that both the algebraic and graphical solutions agree.

Key Concepts

Perimeter Calculation

To find the perimeter of a rectangle, the key is to add up the lengths of all sides. This can be made simpler with the formula for perimeter: \[ P = 2(l + w) \] where \( l \) represents the length and \( w \) the width of the rectangle. By using this formula, you can easily calculate how long the boundary is around the shape. In our exercise, the problem states the perimeter is 54 centimeters. Knowing the relationship between the width and length (length equals twice the width minus 3 cm), we can replace the length in the formula with \( 2w - 3 \). This substitution allows for a straightforward calculation using the perimeter equation without having to first measure each side separately. It also sets the stage for solving the equation to find exact dimensions. Understanding and applying the perimeter formula correctly is crucial for finding the size of any rectangle.

Equation Solving

When it comes to solving an equation, the main goal is to find the value of the variable involved, in this case, the width \( w \). In the previous steps, we derived the equation based on the perimeter formula: \[ 54 = 2((2w - 3) + w) \] The process involves simplifying this equation step by step.

• First, simplify the expression inside the parentheses: \( 3w - 3 \).
• Then, multiply through by 2: \( 6w - 6 \).
• To isolate \( w \), perform the operation of adding 6 to both sides: \( 60 = 6w \).
• Finally, divide both sides by 6 to solve for \( w \), giving us \( w = 10 \) cm.

Solving equations involves balancing and systematically reducing the equation until the variable stands alone. This systematic approach ensures that every solution step you take is valid and logical, ultimately giving the correct width. Once \( w \) is found, calculating the length using our earlier derived expression \( l = 2w - 3 \) completes the quest for dimensions.

Graphical Verification

Graphical verification acts as a secondary check to ensure our calculated solutions are accurate. By plotting the equation representing the perimeter condition, you can visually verify the result.

• Start with the equation: \( 2((2w - 3) + w) = 54 \).
• Graph this equation in terms of \( w \) and observe the solutions on the graph.
• The graph will intersect at the point where both the length and width satisfy the given perimeter condition.
• Check if this point corresponds to \( w = 10 \) and consequently \( l = 17 \).

By visually confirming our solution, this method assures us that the algebraic solution aligns with the graphical representation. It not only verifies correctness but also reinforces the understanding of how algebra and geometry beautifully connect. Graphical verification can be particularly helpful in scenarios where you need to double-check work or explain solutions graphically for a better visual understanding.