To determine the distance between the port and the island, we need to break down the journey into two segments and calculate the resultant displacement vector.
### Step-by-Step Solution:
1. Initial Leg of the Journey:
- The ship initially heads at a [tex]\(135^{\circ}\)[/tex] angle for 400 miles.
- We need to convert this heading into Cartesian coordinates.
- For the [tex]\(135^{\circ}\)[/tex] heading:
[tex]\[
\begin{aligned}
x_1 &= 400 \cdot \cos(135^\circ) \\
y_1 &= 400 \cdot \sin(135^\circ)
\end{aligned}
\][/tex]
- The calculated components are:
[tex]\[
x_1 \approx -282.843 \quad \text{miles (to the left)}
\][/tex]
[tex]\[
y_1 \approx 282.843 \quad \text{miles (upward)}
\][/tex]
2. Second Leg of the Journey:
- The ship then travels at a [tex]\(180^{\circ}\)[/tex] heading for 250 miles.
- We need to convert this heading into Cartesian coordinates.
- For the [tex]\(180^{\circ}\)[/tex] heading:
[tex]\[
\begin{aligned}
x_2 &= 250 \cdot \cos(180^\circ) \\
y_2 &= 250 \cdot \sin(180^\circ)
\end{aligned}
\][/tex]
- The calculated components are:
[tex]\[
x_2 = -250 \quad \text{miles (to the left)}
\][/tex]
[tex]\[
y_2 \approx 0 \quad \text{miles (no change in the y-direction)}
\][/tex]
3. Total Displacement:
- We now add the components of both legs of the journey to find the total displacement.
- Summing up the x-components:
[tex]\[
\text{Total } x = x_1 + x_2 = -282.843 + (-250) = -532.843 \quad \text{miles}
\][/tex]
- Summing up the y-components:
[tex]\[
\text{Total } y = y_1 + y_2 = 282.843 + 0 = 282.843 \quad \text{miles}
\][/tex]
4. Distance from Port to Island:
- Finally, we use the Pythagorean theorem to calculate the straight-line distance from the port to the island.
[tex]\[
\text{Distance} = \sqrt{(\text{Total } x)^2 + (\text{Total } y)^2}
\][/tex]
Substituting the values, we get:
[tex]\[
\text{Distance} \approx \sqrt{(-532.843)^2 + (282.843)^2} \approx 603.26 \quad \text{miles}
\][/tex]
Therefore, the approximate distance between the port and the island is [tex]\(603.26\)[/tex] miles.
