Problem 54 - College Algebra

Question

A man drove 10 \(\mathrm{mi}\) directly east from his home, made a left turn at an intersection, and then traveled 5 \(\mathrm{mi}\) north to his place of work. If a road was made directly from his home to his place of work, what would its distance be to the nearest tenth of a mile?

Step-by-Step Solution

Answer

The direct road would be approximately 11.2 miles long.

Step 1: Identify the Problem

The problem requires us to find the shortest distance between two points - the man's home and his place of work - using direct road construction across a right-angled path.

Step 2: Understand the Path

The man first drives 10 miles east and then 5 miles north, creating a path that forms a right triangle with his home and place of work.

Step 3: Apply the Pythagorean Theorem

To find the direct distance, which is the hypotenuse of the right triangle, apply the Pythagorean Theorem: \[c = \sqrt{a^2 + b^2}\]Where \(a = 10\) miles and \(b = 5\) miles.

Step 4: Calculate the Hypotenuse

Substitute the known values into the Pythagorean Theorem:\[c = \sqrt{10^2 + 5^2} = \sqrt{100 + 25} = \sqrt{125}\]

Step 5: Compute and Round the Result

Calculate the square root of 125 to find:\[\sqrt{125} \approx 11.1803\]Round this to the nearest tenth: 11.2 miles.

Key Concepts

Right Triangle

A right triangle is a fundamental concept in geometry. It consists of a triangle with one angle precisely equal to 90 degrees, which is known as the right angle. In the context of the exercise, the route taken by the man forms a right triangle. Here, his first journey east and then north corresponds to the two sides of the triangle, often referred to as the legs.

Key features of right triangles include:

• The hypotenuse, which is the longest side opposite the right angle. This is the side we seek to find in the problem, as it would represent the direct distance from the man’s home to his workplace.
• The legs, which are the two shorter sides forming the right angle. In this case, one leg measures 10 miles (east direction) and the other 5 miles (north).

Many real-life applications use right triangles to determine shortest distances or solve problems involving angles and dimensions.

Distance Calculation

Distance calculation often involves determining the shortest path between two points. For right triangles, this task can be elegantly handled by the Pythagorean Theorem. This brilliant theorem states that in a right triangle, the square of the hypotenuse (\( c \)) is equal to the sum of the squares of the other two sides (\( a \) and \( b \)).

• In our problem, the direct distance one seeks is equivalent to the hypotenuse of the right triangle formed by the journey.
• This means you can calculate it using the equation: \[ c = \sqrt{a^2 + b^2} \]
• The known values of \( a \) (10 miles) and \( b \) (5 miles), need to simply be placed into this equation to solve for \( c \).

Distance calculation using the Pythagorean Theorem emphasizes the importance of right triangles in determining not just distance, but many aspects of space and structure.

Square Root Calculation

Calculating the square root is a critical mathematical operation when applying the Pythagorean Theorem. Once you have summed the squares of the two legs, finding the hypotenuse requires determining the square root of that value.

• In our scenario, we calculate \( \sqrt{125} \) to find the hypotenuse.
• This results in approximately \( 11.1803 \), and the solution requires rounding to the nearest tenth for a practical result, giving \( 11.2 \) miles as the closest road distance available.
• Rounding is an essential step to deliver a user-friendly outcome without overcomplicating a real-world distance.

Understanding square root calculation helps ensure accuracy when dealing with Pythagorean products, providing precise results suitable for applications like engineering, architecture, and everyday distance measuring.