Problem 33 - Algebra for College Students

Question

Solve each problem. The Bermuda Triangle supposedly causes trouble for aircraft pilots. It has a perimeter of \(3075 \mathrm{mi}\). The shortest side measures 75 mi less than the middle side, and the longest side measures 375 mi more than the middle side. Find the lengths of the three sides.

Step-by-Step Solution

Answer

The lengths are 850 mi, 925 mi, and 1300 mi.

Step 1: Identify Variables

Let's designate the middle side as a variable, say, let it be denoted by \(x.\) The shortest side, therefore, would be \(x - 75\) and the longest side would be \(x + 375.\)

Step 2: Set Up the Equation

According to the problem statement, the perimeter of the Bermuda Triangle is \(3075 \, \mathrm{mi}.\) Therefore, we can set up the equation based on the perimeter: \[ x + (x - 75) + (x + 375) = 3075 \]

Step 3: Simplify the Equation

Combine like terms to simplify the equation: \[ 3x + 300 = 3075 \]

Step 4: Solve for x

Isolate \(x\) by subtracting 300 from both sides: \[ 3x = 2775 \] Then, divide by 3: \[ x = 925 \]

Step 5: Find the Lengths of the Sides

Now that we have \(x = 925,\) we can find the lengths of the three sides: \[ \text{Shortest side} = x - 75 = 925 - 75 = 850 \, \mathrm{mi} \] \[ \text{Middle side} = x = 925 \, \mathrm{mi} \] \[ \text{Longest side} = x + 375 = 925 + 375 = 1300 \, \mathrm{mi} \]

Key Concepts

algebraic equations

Understanding algebra is essential to solving this type of triangle problem. Algebraic equations are mathematical statements that have equal expressions on both sides of an equals sign. To solve an algebraic equation, you often need to find the unknown variable. In our example, solving for the side lengths involves setting up and solving such an equation. Using algebra helps simplify real-world problems and find accurate solutions. To solve the Bermuda Triangle problem, we begin by translating the word problem into a mathematical equation and then using algebra to solve for the unknown variable, which in this case is one of the triangle’s sides.

perimeter

The perimeter of a triangle is the total distance around the triangle. For a triangle, you find the perimeter by adding up the lengths of all three sides. In our Bermuda Triangle problem, the perimeter is given as 3075 miles. Knowing the perimeter helps us set up our equation. By expressing the side lengths in terms of a single variable, we can sum them up and compare the result to the given perimeter. This comparison forms the basis for our equation, which is then solved to find the individual side lengths. Understanding perimeter is crucial for accurately solving geometric problems like this one.

variable identification

Identifying variables correctly is a key step in solving algebraic equations. A variable is a symbol, often a letter, that represents an unknown value. In our problem, we choose to let the middle side be represented by the variable \( x \). This choice simplifies the problem because it allows us to express the other two sides in terms of \( x \). The shortest side becomes \( x - 75 \) and the longest side \( x + 375 \). By identifying and defining the variables this way, the problem becomes much easier to solve. Clearly defining variables helps avoid confusion and makes the mathematical relationships clearer.

problem solving

Solving math problems often involves breaking them down into smaller, manageable steps. Let's look at our triangle problem. First, we identified the variables for the side lengths. Next, we set up an equation that represents the perimeter by summing those sides. Then, we simplified and solved the equation for the variable \( x \). Finally, we calculated the lengths of all sides using the value of \( x \). This systematic approach makes even complex problems more approachable. Successful problem-solving involves understanding the problem, breaking it down into smaller parts, and tackling each part methodically.