Problem 16 - A Graphical Approach to College Algebra

Question

$$\begin{aligned} &\text {Solve each problem analytically, and support your solution}\\\ &\text {graphically.} \end{aligned}$$ Perimeter of a Plot of Land The perimeter of a triangular plot of land is 2400 feet. The longest side is 200 feet less than twice the shortest. The middle side is 200 feet less than the longest side. Find the lengths of the three sides of the triangular plot.

Step-by-Step Solution

Answer

The sides of the triangle are 600 feet, 800 feet, and 1000 feet.

Step 1: Define Variables

Let \( x \) be the length of the shortest side of the triangular plot. Then the longest side can be represented as \( 2x - 200 \) feet because it is 200 feet less than twice the shortest side. The middle side is 200 feet less than the longest side, so it can be written as \( 2x - 400 \) feet.

Step 2: Setup the Perimeter Equation

The perimeter of the triangle is given as 2400 feet. The perimeter is the sum of the lengths of all sides. Thus, we have the equation: \( x + (2x - 200) + (2x - 400) = 2400 \).

Step 3: Simplify the Equation

Combine like terms to simplify the equation: \( x + 2x - 200 + 2x - 400 = 2400 \) becomes \( 5x - 600 = 2400 \).

Step 4: Solve for x

Add 600 to both sides to isolate the term with \( x \): \( 5x = 3000 \). Then divide by 5 to solve for \( x \): \( x = 600 \).

Step 5: Determine Each Side Length

Now that the shortest side is \( x = 600 \) feet, calculate the other sides. The longest side is \( 2x - 200 = 2(600) - 200 = 1000 \) feet. The middle side is \( 2x - 400 = 2(600) - 400 = 800 \) feet.

Step 6: Verify the Solution

Check the perimeter: \( 600 + 800 + 1000 = 2400 \) feet, which matches the given perimeter. Thus, the solution is correct.

Step 7: Support Graphically

To visualize, sketch a triangle on graph paper or using software with sides 600 feet, 800 feet, and 1000 feet, ensuring the sum equals the perimeter to cross-verify it graphically.

Key Concepts

Algebraic Equations

In solving real-world problems such as finding the perimeter of a triangle, algebraic equations are invaluable. An algebraic equation involves a variable, which represents an unknown quantity in the problem. In this exercise, the problem provides the perimeter and relationships between the sides of a triangular plot of land. To express these relationships algebraically, we define a variable.

Let's denote the shortest side as \( x \). Since the longest side is 200 feet less than twice the shortest, we represent it as \( 2x - 200 \). Similarly, the middle side is 200 feet less than the longest side, formulated as \( 2x - 400 \).

These expressions combined in an equation help us calculate the unknown side lengths. By setting up the equation for the perimeter \( x + (2x - 200) + (2x - 400) = 2400 \), we efficiently use algebra to solve for \( x \). This strategy succinctly tackles problems having multiple unknowns, exemplifying the power and simplicity of algebraic equations in problem-solving.

Problem Solving Steps

Effective problem-solving involves a structured approach that simplifies complex tasks into manageable steps. In this exercise, determining the side lengths of a triangle from a given perimeter exemplifies a step-by-step methodology.

Here’s how:

• **Define the Variables:** Establish a base, by setting one side of the triangle to a variable \( x \), then noting relationships among sides.
• **Formulate the Equation:** Use known values and relationships to set up a formal algebraic equation capturing the problem’s essence.
• **Simplify:** Consolidate terms in the equation for clarity, reducing it to an actionable form.
• **Solve:** Algebraically manipulate the equation to find the value of \( x \), ensuring you isolate the variable step by step.
• **Translate Solution:** Convert the solved variable back into real-world side lengths.

Graphical Verification

Graphical verification offers a visual method to validate algebraic solutions. Once we determine the side lengths, representing the triangle graphically helps confirm their correctness in real-world terms.

To verify graphically, one can sketch the triangle on graph paper or use software tools. This sketch aims to conform to the conditions and dimensions calculated:

• Shortest Side: 600 ft
• Middle Side: 800 ft
• Longest Side: 1000 ft

The importance of graphical verification lies in ensuring that the combined lengths visually and graphically satisfy the condition of perimeter, which is 2400 feet. Such visualization helps iron out conceptual errors, providing an intuitive cross-check that complements purely numerical solutions, enhancing overall problem understanding.