Problem 16 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem analytically, and support your solution graphically. 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 the Variables

Let's define the variables for the sides of the triangle. Denote the shortest side as \( x \). According to the problem: 1. The longest side is 200 feet less than twice the shortest side, so it is \( 2x - 200 \). 2. The middle side is 200 feet less than the longest side, so it is \( (2x - 200) - 200 = 2x - 400 \).

Step 2: Write an Equation for the Perimeter

Since the perimeter of the triangle is 2400 feet, we can write the equation for the sum of the three sides:\[ x + (2x - 200) + (2x - 400) = 2400 \]

Step 3: Simplify the Equation

Combine like terms in the equation:\[ x + 2x - 200 + 2x - 400 = 2400 \] This simplifies to:\[ 5x - 600 = 2400 \]

Step 4: Solve for the Shortest Side

Add 600 to both sides of the equation to isolate terms with \( x \):\[ 5x - 600 + 600 = 2400 + 600 \]\[ 5x = 3000 \]Now, divide both sides by 5 to solve for \( x \):\[ x = 600 \]

Step 5: Calculate the Other Side Lengths

Using the value of \( x \) we found:1. The shortest side is \( x = 600 \) feet.2. The longest side is \( 2x - 200 = 2(600) - 200 = 1200 - 200 = 1000 \) feet.3. The middle side is \( 2x - 400 = 2(600) - 400 = 1200 - 400 = 800 \) feet.

Step 6: Verify the Solution

Verify that the sum of the sides equals the perimeter. Adding all the side lengths: \[ 600 + 1000 + 800 = 2400 \] The solution satisfies the condition given in the problem.

Key Concepts

Analytical Solutions

An analytical solution involves breaking down a problem into a set of logical steps to find a mathematical solution. In this triangle perimeter problem, each statement from the problem is interpreted into a mathematical expression. Here, the first step is to define the problem variables such as the shortest, middle, and longest sides of the triangle. By setting the shortest side as \( x \), we use the given relationships to express the other sides in terms of \( x \). This approach turns a verbal problem into an algebraic one, facilitating solving it step by step. Translating the words into equations is a crucial analytical skill that simplifies solving complex problems by providing a clear path towards finding a solution.

Graphical Representation

Graphical representation can be an excellent way to visualize and confirm the solution to problems, especially in geometry. For this triangle problem, a simple sketch can help. First, draw a triangle and label the sides: \( x \), \( 2x-200 \), and \( 2x-400 \). Even though an accurate plot isn't necessary, this labeling helps one to better understand the relationships between the sides. The sketch serves to emphasize how logic and visualization work together in verifying solution steps. Additionally, a graph could illustrate changes in side lengths relative to changes in \( x \), showing how each part fits into the overall equation.

Algebraic Expressions

Algebraic expressions are key to forming the equations needed to solve the triangle perimeter problem. Initially, expressions such as \( 2x-200 \) and \( 2x-400 \) describe each side of the triangle in a way that reflects the conditions given. These expressions are then combined to form the equation \( x + (2x - 200) + (2x - 400) = 2400 \) based on the total perimeter. This equation is simplified by combining like terms to create \( 5x - 600 = 2400 \). Understanding how to manipulate and solve these expressions is fundamental to algebra, allowing for solutions to emerge naturally once the equation is properly set up.

Perimeter Calculation

Calculating the perimeter is the final step to answer the triangle's dimensions problem. Once the equation is simplified to \( 5x = 3000 \), solving for \( x \) gives the shortest side as 600 feet. Using the defined relationships, we find the longest side to be 1000 feet and the middle side to be 800 feet. The verification step involves adding these side lengths together: \( 600 + 1000 + 800 = 2400 \), which matches the given perimeter. This serves as a crucial step to ensure that calculations are correct and conditions of the problem are met. Understanding perimeter calculation is essential, as it is often a foundational step in geometric problem-solving.