Problem 7 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem analytically, and support your solution graphically. The perimeter of a rectangle is 98 centimeters. The width is 19 centimeters. Find the length.

Step-by-Step Solution

Answer

The length of the rectangle is 30 centimeters.

Step 1: Understand the Perimeter Formula

The formula to calculate the perimeter of a rectangle is given by \( P = 2L + 2W \), where \( P \) is the perimeter, \( L \) is the length, and \( W \) is the width.

Step 2: Identify Given Information

From the exercise, we know that the perimeter \( P = 98 \) cm and the width \( W = 19 \) cm. Our task is to find the length \( L \).

Step 3: Substitute Known Values into the Formula

We substitute the known values into the perimeter formula: \( 98 = 2L + 2(19) \). This equation comes from replacing \( W \) with 19 cm and \( P \) with 98 cm.

Step 4: Simplify the Equation

Simplify the equation by calculating \( 2(19) \), which equals 38. The equation now becomes \( 98 = 2L + 38 \).

Step 5: Solve for Length \( L \)

Isolate \( 2L \) by subtracting 38 from both sides of the equation: \( 98 - 38 = 2L \). This simplifies to \( 60 = 2L \). Now, divide both sides by 2 to solve for \( L \): \( L = 30 \).

Step 6: Verify the Solution Graphically

Draw a rectangle with width 19 cm and length 30 cm. Calculate the perimeter graphically to check: \( 2(30) + 2(19) = 98 \) cm, confirming the solution is correct.

Key Concepts

Perimeter Formula

The perimeter is an important concept in geometry, especially for understanding rectangles. In its essence, the perimeter represents the total distance around a geometric shape. The formula for calculating the perimeter of a rectangle is given by: \( P = 2L + 2W \). This equation incorporates two key variables:

• \( L \) - the length of the rectangle
• \( W \) - the width of the rectangle

By summing twice the length and twice the width, you account for all four sides of the rectangle.
Perimeter calculations are useful not only in academic exercises but also in real-world problems related to construction, crafting, and various design projects.
Finding the perimeter helps to allocate resources or understand spatial distributions effectively.

Analytical Solution

An analytical solution involves a logical approach with mathematical calculations to solve a problem systematically. For this exercise:

• Recognize the given information: a rectangle's perimeter is 98 cm, and its width is 19 cm.
• Apply the perimeter formula: \( 98 = 2L + 2(19) \).
• Simplify by calculating \( 2(19) = 38 \), transforming the equation into \( 98 = 2L + 38 \).
• Isolate the variable \( L \) by subtracting 38 from both sides: \( 60 = 2L \).
• Finally, divide by 2 to solve for \( L \), yielding \( L = 30 \) cm.

This direct method ensures that all calculated steps lead to a logical and verifiable solution to the problem. This process enhances understanding of how algebraic manipulations can be applied effectively.

Graphical Representation

Graphical representation aids in visualizing mathematical concepts, making them easier to understand. For the rectangle in this exercise:
Draw a rectangle with a length of 30 cm and width of 19 cm. By plotting this:

• One can easily see how the lengths and widths add up to form the perimeter.
• Visual checks can confirm the formulaic solution derived in the analytical steps.
• It provides an effective visual confirmation that \( 2 \times 30 + 2 \times 19 = 98 \) cm is correct.

Creating a visual representation serves as a powerful tool for learning, helping reinforce the analytical findings and providing greater clarity through a spatial understanding.

Rectangular Dimensions

Rectangular dimensions refer to the length and width of a rectangle, two fundamental elements. These dimensions define the shape and size of a rectangle:

• Length (30 cm in this example) is often considered the longer side of a rectangle.
• Width (19 cm in this exercise) is the shorter side, although these roles can reverse based on context.

Understanding the role and relationship of these dimensions is crucial for perimeter calculations and numerous practical applications.
Knowing the dimensions allows one to plan for space utilization, material usage, and cost estimations in real-world problems, thereby bridging the gap from theoretical exercises to practical applications.