Problem 67 - Prealgebra

Question

A surveying team wants to calculate the length of a straight tunnel through a mountain. They form a right angle by connecting lines from each end of the proposed tunnel. One of the connecting lines is 3 miles, and the other is 4 miles. What is the length of the proposed tunnel?

Step-by-Step Solution

Answer

The length of the tunnel is 5 miles.

Step 1: Identify the Problem

The problem involves finding the length of the hypotenuse of a right triangle. The two given sides of the triangle (3 miles and 4 miles) are the legs.

Step 2: Apply the Pythagorean Theorem

The Pythagorean Theorem helps in calculating the length of the hypotenuse (c) in a right-angled triangle with legs a and b using the formula: \( c = \sqrt{a^2 + b^2} \). Here, let \( a = 3 \) miles and \( b = 4 \) miles.

Step 3: Calculate Each Squared Term

First, we calculate \( 3^2 = 9 \) and \( 4^2 = 16 \).

Step 4: Add the Squared Terms

Combine both square results: \( 9 + 16 = 25 \). This sum represents \( a^2 + b^2 \).

Step 5: Find the Square Root

To find the length of the tunnel, which is the hypotenuse, take the square root of 25, giving us \( \sqrt{25} = 5 \).

Step 6: Conclusion

The length of the tunnel is calculated to be 5 miles, which is the length of the hypotenuse.

Key Concepts

Understanding Right Triangles

In geometry, a right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This 90-degree angle is what makes it special and gives it the name "right triangle." There are a few important parts to know about right triangles:

• Legs: The two sides that form the 90-degree angle are called the legs. In our example, these are the lines that are 3 miles and 4 miles long.


• Hypotenuse: This is the longest side of the right triangle, opposite the 90-degree angle. It's the side we want to find in this problem.

Right triangles often appear in practical problems, like building designs, navigation, and, as in this problem, construction planning. Recognizing the right triangle can be the first step in solving many real-world problems effectively.

Hypotenuse Calculation using the Pythagorean Theorem

To find the hypotenuse in a right triangle, we use a very handy tool called the Pythagorean Theorem. This theorem, discovered by the ancient Greek mathematician Pythagoras, gives us a way to calculate the longest side of a right triangle (the hypotenuse) when the lengths of the other two sides are known.

• The Formula: The theorem is expressed as: \[ c = \sqrt{a^2 + b^2} \] where \( c \) is the hypotenuse, and \( a \) and \( b \) are the legs.


• Given Values: In this specific problem, \( a = 3 \) miles and \( b = 4 \) miles are the lengths of the two legs.


• Applying the Theorem: We substitute the values into the formula: \[ c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \] miles.

Using the Pythagorean Theorem reveals that the length of the tunnel (hypotenuse) is 5 miles.

Step-by-Step Problem Solving Process

Breaking a problem into smaller, more manageable steps can be highly beneficial for understanding and solving it. Here is how we used a step-by-step approach to tackle this exercise:

• Identify: First, recognize the nature of the problem — calculating the hypotenuse of a right triangle.


• Apply: Use the Pythagorean Theorem to see how the known values of the legs relate to the unknown hypotenuse.


• Calculate:
  • Find the square of each leg: \(3^2 = 9\) and \(4^2 = 16\).
  
  
  • Add these squared terms to get the result for the equation: \(9 + 16 = 25\).
  
  
  • Take the square root of the sum: \(\sqrt{25} = 5\) miles.
  
  
  This systematic approach ensures that nothing gets overlooked, and each part of the process builds on the previous one, leading to an accurate solution.