Problem 2 - Basic Engineering Mathematics

Question

Two aircraft leave an airfield at the same time. One travels due north at an average speed of \(300 \mathrm{~km} / \mathrm{h}\) and the other due west at an average speed of \(220 \mathrm{~km} / \mathrm{h}\). Calculate their distance apart after fourhours.

Step-by-Step Solution

Answer

The distance between the aircraft is approximately 1488.1 km.

Step 1: Calculate Distance Travelled by Each Aircraft

To find out how far each aircraft has traveled, use the formula for distance, which is \( \text{Distance} = \text{Speed} \times \text{Time} \). For the northbound aircraft: \( 300 \text{ km/h} \times 4 \text{ hours} = 1200 \text{ km} \). For the westbound aircraft: \( 220 \text{ km/h} \times 4 \text{ hours} = 880 \text{ km} \).

Step 2: Set Up the Right Triangle

Since one aircraft is traveling north and the other west, they form a right triangle with their paths. The distance the northbound aircraft traveled is one leg, and the distance the westbound aircraft traveled is the other leg.

Step 3: Apply the Pythagorean Theorem

To find the hypotenuse of the right triangle, which is the distance between the two aircraft, use the Pythagorean theorem: \( c^2 = a^2 + b^2 \). Substitute \( a = 1200 \text{ km} \) and \( b = 880 \text{ km} \): \[ c^2 = 1200^2 + 880^2 \].

Step 4: Calculate and Solve for c

Compute the squares: \( 1200^2 = 1440000 \) and \( 880^2 = 774400 \). Add these results: \( 1440000 + 774400 = 2214400 \). Take the square root of the sum to find \( c \): \( c = \sqrt{2214400} \approx 1488.1 \text{ km} \).

Key Concepts

Distance Calculation

To calculate the distance traveled by an object, we can use a simple formula derived from basic kinematics. This formula is:

• Distance = Speed × Time

In the given exercise, two aircraft are traveling at constant speeds. The northbound aircraft travels at 300 km/h and the westbound one at 220 km/h. Both are flying for four hours. By using the distance formula:

• The northbound aircraft covers: \(300 \text{ km/h} \times 4 \text{ hours} = 1200 \text{ km}\).
• The westbound aircraft covers: \(220 \text{ km/h} \times 4 \text{ hours} = 880 \text{ km}\).

This calculation helps set up the following problem, where these distances form the legs of a right triangle.

Right Triangle

In mathematics, a right triangle is a triangle where one of the angles is exactly 90 degrees. For such triangles, there is a special relationship between the lengths of the sides, known as the Pythagorean Theorem. In this exercise, the path of each aircraft travels forms the two legs of a right triangle because they are moving perpendicularly: one north, the other west. The two distances form the legs of the triangle:

• Northbound leg: 1200 km.
• Westbound leg: 880 km.

The hypotenuse of this triangle is the direct distance between the two planes. Understanding how these directional distances relate geometrically through a triangle is crucial to solving the problem.

Average Speed

Average speed is a concept often used in physics and mathematics to determine how fast an object is moving over a certain period. It is calculated by dividing the total distance traveled by the total time taken. For constant speeds, like in the case of the aircraft, the average speed equals their constant traveling speed. In this problem:

• The northbound aircraft's average speed is 300 km/h.
• The westbound aircraft's average speed is 220 km/h.

These figures are critical as they help determine the distance each aircraft covers, aiding in further calculations regarding their positions after four hours. Understanding average speed and its application in distance calculation is foundational for solving many motion-related problems.

Mathematical Problem Solving

Mathematical problem solving involves breaking down a complex problem into more manageable parts, using systematic methods or mathematical principles. In this exercise, solving the problem involves:

• Identifying the problem: Calculate the distance between the two aircraft.
• Breaking it into smaller parts: Firstly, calculate the distance covered by each aircraft.
• Applying mathematical concepts: Use the Pythagorean Theorem to calculate the direct distance, i.e., the hypotenuse of the triangle formed.
• Executing calculations: Compute squares and square roots to solve for the distance.

By approaching problems methodically and using known formulas like distance calculation and the Pythagorean theorem, complex tasks become simpler and more intuitive.