Sure, let’s work through the problem step by step.
1. Understanding Directions and Angles:
- The cruise ship leaves its port at a heading of [tex]\(135^\circ\)[/tex].
- Then it travels for 400 miles and changes its heading to [tex]\(180^\circ\)[/tex].
- Finally, it travels for another 250 miles to reach the island.
2. Angles Between Courses:
- Heading [tex]\(135^\circ\)[/tex] is measured from the north in the clockwise direction.
- Heading [tex]\(180^\circ\)[/tex] is due south.
Therefore, the turn made from [tex]\(135^\circ\)[/tex] to [tex]\(180^\circ\)[/tex] results in a change in direction.
To find the angle between the two legs of the journey, we calculate the difference:
[tex]\[
\text{Angle between directions} = 180^\circ - 135^\circ = 45^\circ
\][/tex]
3. Using the Law of Cosines:
The Law of Cosines states:
[tex]\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(\theta)
\][/tex]
where:
- [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are the sides (400 miles and 250 miles, respectively),
- [tex]\(\theta\)[/tex] is the angle between these sides (45 degrees or in radians, [tex]\(\frac{\pi}{4}\)[/tex]),
- [tex]\(c\)[/tex] is the distance from the port to the island.
4. Converting the Angle to Radians:
[tex]\[
45^\circ = \frac{\pi}{4} \text{ radians}
\][/tex]
5. Calculating the Distance [tex]\(c\)[/tex]:
Plugging the values into the Law of Cosines formula:
[tex]\[
c^2 = 400^2 + 250^2 - 2 \times 400 \times 250 \times \cos\left(\frac{\pi}{4}\right)
\][/tex]
We know that [tex]\(\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\)[/tex]:
[tex]\[
c^2 = 160000 + 62500 - 2 \times 400 \times 250 \times \frac{\sqrt{2}}{2}
\][/tex]
Simplify the calculations:
[tex]\[
c^2 = 160000 + 62500 - 400 \times 250 \times \sqrt{2}
\][/tex]
[tex]\[
c^2 = 160000 + 62500 - 100000 \sqrt{2}
\][/tex]
6. Finding the Approximate Value:
Summing these together:
[tex]\[
c \approx 284.74 \text{ miles}
\][/tex]
Therefore, the approximate distance between the port and the island is 284.74 miles.
