How Far is the Car from the Starting Point After Its Journey?

In this article, we will explore a common geometry problem involving a car traveling in different directions and calculate the distance from its final position to its initial starting point. We will use a coordinate system to break down the movements and apply the distance formula to find the exact distance. Additionally, we will consider different scenarios to understand the impact of starting points on the car's journey.

Initial Setup and Movements

Let's assume the car starts at the origin point (0, 0).

The car's movements are as follows:

Travels 5 miles north, so the new position is (0, 5). Travels 6 miles west, so the new position is (-6, 5). Travels 8 miles south, so the new position is (-6, -3).

Using the Distance Formula

To find the distance from the car's final position (-6, -3) back to the starting point (0, 0), we use the distance formula:

Distance √[(x2 - x1)2 (y2 - y1)2]

Substituting the coordinates:

Distance √[(-6 - 0)2 (-3 - 0)2] √[36 9] √45 3√5

Therefore, the car is approximately 6.71 miles from the initial starting point.

Special Scenarios

Let's consider some special scenarios to understand how the starting point affects the car's final distance from the starting point.

Starting Outside the Polar Region

Assume the car starts three miles north of the south pole. In this case, the car's movements would bring it directly to the south pole but still be 3 miles from the starting point. Therefore, the car is 3 miles from the starting point.

Round Trip Consideration

If we assume that the car is on a round trip, the journey would be:

Travels 5 miles north. Travels 6 miles west. Travels 8 miles south (5 miles off). Needs to travel 3 miles east to return to the starting point.

However, since the car has already moved 6 miles west, it must travel an additional 6 miles east to return to the starting point. Thus, the total distance from the starting point is:

Distance 3 miles (south) 6 miles (west travel) 9 miles.

Conclusion

The final distance of the car from the starting point depends on the initial conditions. In the standard scenario, the distance is approximately 6.71 miles. In a scenario where the car starts three miles north of the south pole, the distance is 3 miles. And for a round trip scenario, the distance is 9 miles.

Understanding these scenarios can greatly enhance your problem-solving skills in real-world situations involving navigation and positioning.