Problem 13 - Precalculus Student Solutions Manual 5th

Question

Solve each problem. Perimeter of a Plot of Land The perimeter of a triangular plot of land is \(2400 \mathrm{ft}\). The longest side is 200 ft less than twice the shortest. The middle side is 200 ft less than the longest side. Find the lengths of the three sides of the triangular plot.

Step-by-Step Solution

Answer

The sides are 600 ft, 800 ft, and 1000 ft.

Step 1: Define the variables

Let the length of the shortest side be represented by \(x\).

Step 2: Express the other sides in terms of x

The longest side is 200 ft less than twice the shortest side. Thus, it is represented by \(2x - 200\). The middle side is 200 ft less than the longest side, which can be represented by \(2x - 200 - 200 = 2x - 400\).

Step 3: Set up the equation using the perimeter

The perimeter of the triangle is the sum of all three sides: \[ x + (2x - 200) + (2x - 400) = 2400 \]

Step 4: Simplify and solve for x

Combine the terms: \[ x + 2x - 200 + 2x - 400 = 2400 \] \[ 5x - 600 = 2400 \] \[ 5x = 3000 \] \[ x = 600 \]

Step 5: Find the lengths of all three sides

Using \(x = 600\), the shortest side is \(600 \text{ ft}\). The longest side is \[ 2(600) - 200 = 1000 \text{ ft} \]. The middle side is \[ 1000 - 200 = 800 \text{ ft} \]. Therefore, the sides are 600 ft, 800 ft, and 1000 ft.

Key Concepts

Solving Equations

To find the lengths of the sides of the triangular plot, it's crucial to solve the equation that represents the perimeter. Equations help us find unknown values by balancing both sides.
Let's start with the equation for the perimeter of the triangle:

1. The perimeter, which is the sum of all sides, is given as 2400 ft.
2. Make an equation by summing up the shortest side (\(x\))
   and the expressions of the other sides:
• The longest side is \(2x - 200\).
• The middle side is \(2x - 400\).
3. Write it all together: \(x + (2x - 200) + (2x - 400) = 2400\).

From here, combining like terms to simplify the equation is necessary. Simply follow these steps diligently:

• Combine \(x\) terms: \(x + 2x + 2x\) becomes \(5x\).
• Combine constants: \(-200 - 400\) becomes \(-600\).
• Now, we have \(5x - 600 = 2400\).

Solving for \(x\):

• Add \(600\) to both sides to get \(5x = 3000\).
• Finally, divide by \(5\): \(x = 600\).

With \(x = 600\), you can find the lengths of the sides by plugging it back into the expressions provided.

Variable Representation

When tackling algebra problems, defining your variables is your first step. In this problem, we define \(x\) to represent the shortest side of the triangle.
After defining \(x\), we express other sides using this variable:

1. \(x:\) The shortest side.
2. \(2x - 200:\) The longest side, which is 200 ft less than twice the shortest side.
3. \(2x - 400:\) The middle side, 200 ft less than the longest side.

Variable representation is crucial because it allows us to translate the verbal problem into algebraic expressions. These expressions can then be used in equations to solve for unknowns. Using variables also ensures clarity and consistency in our solutions.
Break it down:

• Identify what needs to be found. Here, the side lengths.
• Use variables to represent the unknown values, e.g., \(x\) for the shortest side.
• Translate conditions into algebraic terms, e.g., \(2x - 200\) for the longest side.

This consistent approach will make solving any problem more manageable.

Geometric Properties

Understanding geometric properties is vital when solving problems regarding shapes, like triangles. For triangles, the perimeter is the sum of all its sides. This basic property underlies the given problem.
Here’s how these properties help:

• Perimeter Sum: Adding the three sides gives us the total perimeter of the triangle, \(2400\) ft.
• Triangle Sides Relations: The triangle's sides are related by specific conditions given in the problem, like '200 ft less' or 'twice the shortest side'.
• Using Geometric Shapes: Represent each side as an algebraic expression, ensuring the sum equals the given perimeter.

Applying these properties:

1. Set up an equation representing the perimeter.
2. Simplify this equation to find the variable \(x\).
3. Use this variable to determine the specific lengths of the sides.

For instance, with \(x = 600\):

• The shortest side \(600\) ft.
• The longest side: \(2(600) - 200 = 1000\) ft.
• The middle side: \(1000 - 200 = 800\) ft.

Understanding these properties ensures a clear path for problem solving.