Understanding the Auto Trip Problem

The problem asks us to find out how much less distance Sita would have travelled if the auto driver had taken the direct route from her home in Andheri (A) to her college in Fatehpuri (F), compared to the circuitous route he actually took.

We need to calculate two things:

1. The total distance covered by the auto driver on the indirect route.
2. The direct distance between Andheri (A) and Fatehpuri (F).

The difference between the indirect distance and the direct distance will give us the distance saved.

Calculating the Indirect Distance of the Auto Trip

The auto driver took the following path with specified distances between points:

• Andheri (A) to Bandra (B): 10 km
• Bandra (B) to Colaba (C): 8 km
• Colaba (C) to Dalhousi Building (D): 12 km
• Dalhousi Building (D) to Elphinston point (E): 8 km
• Elphinston point (E) to Fatehpuri (F): 4 km

The total distance of the indirect route is the sum of all these segments:

Indirect Distance \( = AB + BC + CD + DE + EF \)

Indirect Distance \( = 10 \text{ km} + 8 \text{ km} + 12 \text{ km} + 8 \text{ km} + 4 \text{ km} \)

Indirect Distance \( = 42 \text{ km} \)

So, Sita actually travelled 42 km.

Determining the Direct Distance from Andheri to Fatehpuri

The problem states the direct road had no turns. Let's analyze the turns taken by the auto driver to understand the relative positions of the points and the path.

Starting from A, the driver goes 10 km to B.

At B, the driver turns right and goes 8 km to C.

At C, the driver turns left and goes 12 km to D.

At D, the driver turns left and goes 8 km to E.

At E, the driver turns right and goes 4 km to F.

Let's visualize this path, assuming the initial direction from A to B is North for simplicity:

• A to B: 10 km North
• B to C: Turn right (East), 8 km East
• C to D: Turn left (North), 12 km North
• D to E: Turn left (West), 8 km West
• E to F: Turn right (North), 4 km North

Relative displacement from A to F:

• Northward movement: 10 km (A to B) + 12 km (C to D) + 4 km (E to F) \( = 26 \text{ km} \) North
• Eastward movement: 8 km (B to C) East
• Westward movement: 8 km (D to E) West

The net East-West displacement is 8 km East - 8 km West \( = 0 \text{ km} \). This means point F is directly North of point A.

The total Northward displacement is 26 km. Since there is no net East-West displacement, the direct distance from A to F is simply the total Northward displacement.

Direct Distance \( = 26 \text{ km} \)

This confirms that the direct route between Andheri and Fatehpuri was 26 km long.

Finding the Distance Saved on the Direct Route

The distance saved is the difference between the distance travelled on the indirect route and the distance on the direct route.

Distance Saved \( = \text{Indirect Distance} - \text{Direct Distance} \)

Distance Saved \( = 42 \text{ km} - 26 \text{ km} \)

Distance Saved \( = 16 \text{ km} \)

Therefore, Sita would have travelled 16 km less if the auto driver had taken the direct route.

The difference is \(42 - 26 = 16\) km.

Revision Table: Auto Trip Distance Calculation

Additional Information: Understanding Displacement vs. Distance

This problem highlights the difference between total distance travelled and displacement.

• Distance is the total length of the path covered. In this case, the auto travelled 42 km along the winding road.
• Displacement is the straight-line distance between the starting point and the ending point, along with direction. It only depends on the initial and final positions, not the path taken. In this case, the displacement from A to F is 26 km North. The direct route distance is the magnitude of this displacement.

Even though the auto driver made several turns, the net effect of moving 8km East and then 8km West is that there was no overall change in the East-West position relative to the starting point after those legs. The net movement was entirely in the North-South direction, adding up the distances from the Northward segments.