Question

A car travels 4.8km east, then 2.1km north, and finally 1.9km west. Assuming a flat rectangular world, what is the shortest distance the car would have traveled if it went directly from the starting point to the final point?

Answer 1

To determine the shortest distance the car would have traveled directly from the starting point to the final point, we can first find the car's net position after its journey.

1. **Starting Point:** Let's assume the car starts at the origin, point \( A(0, 0) \).

2. **First Leg:** The car travels 4.8 km east.
- New position after this leg: \( B(4.8, 0) \).

3. **Second Leg:** Then, the car travels 2.1 km north.
- New position after this leg: \( C(4.8, 2.1) \).

4. **Third Leg:** Finally, the car travels 1.9 km west.
- The new position is calculated as follows:
\[
x \text{-coordinate} = 4.8 - 1.9 = 2.9
\]
- Therefore, the final position of the car is \( D(2.9, 2.1) \).

Now, we need to calculate the direct distance from the starting point \( A(0, 0) \) to the final point \( D(2.9, 2.1) \).

5. **Distance Formula:** The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
\]
Applying this formula to points \( A(0, 0) \) and \( D(2.9, 2.1) \):
\[
d = \sqrt{(2.9 - 0)^2 + (2.1 - 0)^2}
\]
\[
= \sqrt{(2.9)^2 + (2.1)^2}
\]
\[
= \sqrt{8.41 + 4.41}
\]
\[
= \sqrt{12.82}.
\]

6. **Calculating the Square Root:**
\[
\sqrt{12.82} \approx 3.58 \text{ km (to 2 decimal places)}.
\]

Thus, the shortest distance the car would have traveled if it went directly from the starting point to the final point is approximately **3.58 km**.