Question

A park has two gates, Gate 1 and Gate 2. These two gates are connected via two alternate paths. If one takes the first path from Gate 1, they need to walk 80m towards east, then 80m towards south, and finally 20m towards west to arrive at Gate 2. The second path is a semi-circle connecting the two gates, where the diameter of the semi-circle is the straight-line distance between the two gates.
A person walking at a constant speed of 5 kilometers/hour enters the park through Gate 1, walks along the first path to reach Gate 2 and then takes the second path to come back to Gate 1.
Which of the following is the CLOSEST to the time the person takes, from entering the park to coming back to Gate 1, if she never stops in between?

• A. 6 minutes
• B. 5 minutes 30 seconds
• C. 3 minutes 30 seconds
• D. 4 minutes 30 seconds
• E. 4 minutes

Hint:

For problems involving circular paths or semi-circles, use the Pythagorean theorem to calculate the straight-line distance (diameter), then find the perimeter of the semi-circle to get the distance.

Answer 1

The Correct Option is B

Step 1: Understand the problem setup.
We are given two paths. Let's break the problem down: - First Path (Gate 1 to Gate 2): - Walk 80m towards east, - Then 80m towards south, - Finally 20m towards west. The total distance covered in the first path is: \[ 80 + 80 + 20 = 180 \, \text{meters} \] - Second Path (Gate 2 to Gate 1): The second path is a semi-circle connecting the two gates. The diameter of the semi-circle is the straight-line distance between Gate 1 and Gate 2.
Step 2: Calculate the straight-line distance between Gate 1 and Gate 2.
The straight-line distance between Gate 1 and Gate 2 is the hypotenuse of a right triangle formed by the 80m eastward and 80m southward sides. Using the Pythagorean theorem: \[ \text{Hypotenuse} = \sqrt{80^2 + 80^2} = \sqrt{6400 + 6400} = \sqrt{12800} = 113.14 \, \text{meters} \] This is the diameter of the semi-circle.
Step 3: Calculate the distance along the semi-circle.
The circumference of the full circle would be \( \pi \times 113.14 \), but since we are only dealing with half of the circle (the semi-circle), the distance covered in the second path is: \[ \text{Distance along the semi-circle} = \frac{\pi \times 113.14}{2} \approx 177.47 \, \text{meters} \]
Step 4: Total distance covered.
The total distance covered by the person is the sum of the distances along the two paths: \[ \text{Total distance} = 180 \, \text{meters} + 177.47 \, \text{meters} = 357.47 \, \text{meters} \]
Step 5: Calculate the time taken.
The person walks at a speed of 5 kilometers/hour, which is equivalent to \( \frac{5000 \, \text{meters}}{3600 \, \text{seconds}} \approx 1.3889 \, \text{m/s} \). The time taken to cover the total distance is: \[ \text{Time} = \frac{357.47 \, \text{meters}}{1.3889 \, \text{m/s}} \approx 257.5 \, \text{seconds} \] Converting seconds into minutes: \[ 257.5 \, \text{seconds} \approx 4 \, \text{minutes} 17.5 \, \text{seconds} \] Thus, the closest time is 5 minutes 30 seconds.