Step-by-Step Displacement Analysis

To find the final distance from home, we can track Amit's net movement in the North-South and East-West directions.

1. Tracing the Path

Let's break down the journey step by step, starting from home (the origin):

1. 15 km Southwards: Net displacement is 15 km South.
2. Turn right (now facing West) and cycle 7 km: Net displacement is 15 km South and 7 km West.
3. Turn right (now facing North) and cycle 10 km: The 10 km North movement partially cancels the 15 km South movement.
   Net Southward displacement = 15 - 10 = 5 km.
   Net displacement is now 5 km South and 7 km West.
4. Turn left (now facing West) and cycle 5 km: This adds to the Westward displacement.
   Net Westward displacement = 7 + 5 = 12 km.
   Final net displacement is 12 km West and 5 km South.

2. Calculating the Final Distance

Amit's final position is 12 km to the West and 5 km to the South of his home. This forms the two legs of a right-angled triangle, with the direct distance back home being the hypotenuse.

We can use the Pythagorean theorem \(a^2 + b^2 = c^2\) to find this distance.

Distance² = (12 km)² + (5 km)²

Distance² = 144 + 25

Distance² = 169

Distance = √169

Distance = 13 km

Conclusion

Amit will have to cycle 13 km to reach his home.