Finding the Shortest Distance Between Two Points

This problem requires us to find the shortest distance between two points, the ice-cream parlor and the central park, given a sequence of movements in different directions. This type of problem can be solved by understanding the concept of displacement and using geometric principles, specifically the Pythagorean theorem.

Analyzing Katrina's Movement

Let's break down Katrina's walk step-by-step from the school:

• Step 1: School to Ice-Cream Parlor: She walked 40 m towards South from her school to reach the ice-cream parlor.
• Step 2: Ice-Cream Parlor to Bus Stop: From the ice-cream parlor, she took a left turn and walked 15 m to reach a bus stop. When facing South, a left turn is towards the East. So, she walked 15 m East from the ice-cream parlor.
• Step 3: Bus Stop to Central Park: From the bus stop, she then took another left turn and walked for 30 m to reach the central park. From the bus stop, she was facing East. A left turn from East is towards the North. So, she walked 30 m North from the bus stop.

Determining the Displacement from Ice-Cream Parlor to Central Park

We need to find the shortest distance between the ice-cream parlor and the central park. Let's consider the ice-cream parlor as our starting point for this specific calculation. From the ice-cream parlor:

• Katrina first moved 15 m East (to the bus stop).
• Then, from the bus stop, she moved 30 m North (to the central park).

These two movements, 15 m East and 30 m North, represent the net displacement from the ice-cream parlor to the central park. These two directions (East and North) are perpendicular to each other. Therefore, the path from the ice-cream parlor directly to the central park forms the hypotenuse of a right-angled triangle, where the two perpendicular sides are 15 m (East displacement) and 30 m (North displacement).

Applying the Pythagorean Theorem

The shortest distance between the ice-cream parlor and the central park is the length of the hypotenuse of the right triangle formed by the displacements. According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse (\(c\)) is equal to the sum of the squares of the other two sides (\(a\) and \(b\)).

The formula is: \(c^2 = a^2 + b^2\)

In our case:

• Side \(a\) = 15 m (East displacement)
• Side \(b\) = 30 m (North displacement)
• Side \(c\) = Shortest distance (hypotenuse)

Calculating the Shortest Distance

Let's plug the values into the Pythagorean theorem:

\(c^2 = (15 \, \text{m})^2 + (30 \, \text{m})^2\)

\(c^2 = 225 \, \text{m}^2 + 900 \, \text{m}^2\)

\(c^2 = 1125 \, \text{m}^2\)

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

\(c = \sqrt{1125 \, \text{m}^2}\)

\(c \approx 33.54 \, \text{m}\)

The approximate shortest distance between the ice-cream parlor and the central park is 33.54 m.

Comparing with Options

Let's look at the given options:

• 37.5 m
• 33.5 m
• 46 m
• 50 m

Our calculated distance, approximately 33.54 m, is closest to 33.5 m.

Therefore, the approximate shortest distance between the ice-cream parlor and the central park is 33.5 m.

Revision Table: Key Movements and Displacements

Additional Information: Distance vs. Displacement

It's important to distinguish between distance and displacement in problems like this:

• Distance: This is the total length of the path traveled. In this problem, the total distance Katrina walked from school to the central park is 40 m (South) + 15 m (East) + 30 m (North) = 85 m. However, the question asks for the shortest distance between two specific points, which is related to displacement.
• Displacement: This is the shortest distance between the starting point and the ending point, along with the direction. It is a vector quantity. The shortest distance is always a straight line. In our calculation, we found the magnitude of the displacement vector from the ice-cream parlor to the central park using the components of movement (15m East and 30m North).

The shortest distance between two points is the magnitude of the displacement vector connecting those points. The Pythagorean theorem is a fundamental tool for calculating this magnitude when the displacement can be broken down into perpendicular components.