Calculating Shortest Distance Between Office and Bank

The problem asks us to find the shortest distance between Shweta's starting point (office) and her ending point (bank) after a series of movements. This involves finding the net displacement, which is the straight-line distance from the start to the end point, regardless of the path taken.

Analyzing Shweta's Movements

Let's break down Shweta's walk step by step:

• Starts at the office.
• Walks 150 m towards the South.
• Turns Right (from facing South, Right is West) and walks 80 m.
• Turns Left (from facing West, Left is South) and walks 60 m.
• Finally turns Left (from facing South, Left is East) and walks 280 m to reach the bank.

Determining Net Displacement

We can represent the movements in terms of East-West and North-South directions to find the total change in position from the office.

Southward movements:

• 150 m South
• 60 m South

Total displacement towards South = $150 \text{ m} + 60 \text{ m} = 210 \text{ m South}$

Eastward/Westward movements:

• 80 m West
• 280 m East

Total displacement towards East = $280 \text{ m East} - 80 \text{ m West} = 200 \text{ m East}$ (since East > West)

So, the final position (bank) is 210 m South and 200 m East from the starting position (office).

Using the Pythagorean Theorem

The office, the bank, and a point representing the net displacement towards South and East form a right-angled triangle. The shortest distance between the office and the bank is the hypotenuse of this triangle. We can use the Pythagorean theorem to find this distance.

Let:

• Net Southward displacement = Perpendicular side = 210 m
• Net Eastward displacement = Base side = 200 m
• Shortest distance (Office to Bank) = Hypotenuse = $d$

According to the Pythagorean theorem: $Hypotenuse^2 = Base^2 + Perpendicular^2$

So, $d^2 = (200 \text{ m})^2 + (210 \text{ m})^2$

Calculating the squares:

$d^2 = 40000 + 44100$

$d^2 = 84100$

To find $d$, we take the square root of both sides:

$d = \sqrt{84100}$

$d = 290 \text{ m}$

The shortest distance between Shweta's office and the bank is 290 m.

Summary of Movements and Displacement

The net displacement is 210 m South and 200 m East. Using Pythagorean theorem: $\sqrt{(-210)^2 + (200)^2} = \sqrt{44100 + 40000} = \sqrt{84100} = 290$ m.

Final Answer

The shortest distance between her office and the bank is 290 m.

Revision Table: Distance and Displacement

Additional Information: Directions and Turns

Understanding directions and turns is crucial for solving navigation problems like this. Standard directions are North, South, East, West.

• When facing North, a Right turn is East, a Left turn is West.
• When facing South, a Right turn is West, a Left turn is East.
• When facing East, a Right turn is South, a Left turn is North.
• When facing West, a Right turn is North, a Left turn is South.

In this problem:

• Starts walking South.
• Turns Right: Now facing West.
• Turns Left: Now facing South again.
• Turns Left: Now facing East.

This detailed breakdown of movements helps confirm the directional components used in the displacement calculation.