Understanding the Journey and Displacement

This problem asks for the shortest distance between a starting point and a final destination after a series of movements in different directions. This type of problem involves calculating the net displacement, which is a vector quantity representing the straight-line distance and direction from the start to the end point.

We can solve this by considering the movements along the East-West axis and the North-South axis separately. Let's assume the starting point is the origin (0,0) of a coordinate system, where East is the positive x-direction and North is the positive y-direction. Consequently, West is the negative x-direction and South is the negative y-direction.

Step-by-Step Movement Analysis

Let's track the man's position based on the given movements:

• Initial Position: Start at (0,0).
• Movement 1: Travels 15 km towards the east.
  Position becomes (0 + 15, 0) = (15, 0).
  Net Eastward movement: 15 km. Net Southward movement: 0 km.
• Movement 2: Turns right and travels 20 km. When traveling East, a right turn is towards the South.
  Position becomes (15, 0 - 20) = (15, -20).
  Net Eastward movement: 15 km. Net Southward movement: 20 km.
• Movement 3: He then turns left and travels 30 km. When traveling South, a left turn is towards the East.
  Position becomes (15 + 30, -20) = (45, -20).
  Net Eastward movement: 15 km + 30 km = 45 km. Net Southward movement: 20 km.
• Movement 4: Finally, he takes a right turn and covers 40 km. When traveling East, a right turn is towards the South.
  Position becomes (45, -20 - 40) = (45, -60).
  Net Eastward movement: 45 km. Net Southward movement: 20 km + 40 km = 60 km.

Calculating Net Displacement

After all the movements, the final position relative to the starting point is (45, -60). This means the man is 45 km to the East and 60 km to the South of his starting point.

The total displacement is a vector from (0,0) to (45, -60). The shortest distance is the magnitude of this displacement vector, which is the straight-line distance between the start and end points.

Using the Pythagorean Theorem for Shortest Distance

The net movement forms two perpendicular legs of a right-angled triangle: 45 km East and 60 km South. The shortest distance between the start and end point is the hypotenuse of this right triangle.

Let \(d\) be the shortest distance. According to the Pythagorean theorem:

\(d^2 = (\text{Net East/West Displacement})^2 + (\text{Net North/South Displacement})^2\)

In this case:

\(d^2 = (45 \text{ km})^2 + (60 \text{ km})^2\)

Now, let's calculate the squares:

\(45^2 = 45 \times 45 = 2025\)

\(60^2 = 60 \times 60 = 3600\)

Substitute these values back into the equation:

\(d^2 = 2025 + 3600\)

\(d^2 = 5625\)

To find \(d\), we take the square root of both sides:

\(d = \sqrt{5625}\)

Calculating the square root:

\(\sqrt{5625} = 75\)

So, the shortest distance between the starting point and the destination is 75 km.

Summary of Movements and Displacement

Total Horizontal Displacement = \(+15 + 0 + 30 + 0 = +45\) km (East)

Total Vertical Displacement = \(0 - 20 + 0 - 40 = -60\) km (South)

Shortest distance \(d = \sqrt{(+45)^2 + (-60)^2} = \sqrt{45^2 + 60^2} = \sqrt{2025 + 3600} = \sqrt{5625} = 75\) km.

Revision Table: Distance and Displacement Concepts

Additional Information on Direction and Distance Problems

Direction and distance problems are common in reasoning and quantitative aptitude tests. They often involve visualizing movement on a 2D plane. Here are some key points:

• Cardinal Directions: North, South, East, West. Remember that North is opposite to South and East is opposite to West.
• Turns: "Turns right" and "Turns left" depend on the current direction of movement. From North, right is East, left is West. From South, right is West, left is East. From East, right is South, left is North. From West, right is North, left is South. A right turn is always 90 degrees clockwise, and a left turn is always 90 degrees counter-clockwise unless specified otherwise.
• Shortest Distance: Always refers to the straight-line distance (displacement) between the two points. This usually involves using the Pythagorean theorem if the net movement results in perpendicular components (like net East/West and net North/South).
• Coordinate System: Using a coordinate system (x, y) with the start at (0,0) and assigning directions to axes (e.g., East=+x, North=+y) simplifies tracking movements and calculating net displacement.
• Pythagorean Triples: Recognizing common Pythagorean triples (like 3-4-5, 5-12-13) or their multiples can speed up calculations (e.g., 45, 60, x is 15 times the 3-4-5 triple, so x is 15 * 5 = 75).