Question

a cruise ship leaves its port at a heading of 135°. it travels for 400 miles, then turns to a heading of 180°. after another 250 miles, it reaches an island. what is the approximate distance between the port and the island? a. 471.7 miles b. 603.3 miles c. 541.5 miles d. 284.7 miles

Explanation:

Step1: Identify the law to use

We can use the Law of Cosines. The Law of Cosines formula for a triangle with sides \(a\), \(b\), \(c\) and the angle \(\theta\) opposite to side \(c\) is \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\). Here, let \(a = 400\), \(b=250\), and \(\theta = 135^{\circ}\), and we want to find the distance \(d\) (let \(d = c\)) between the port and the island.

Step2: Calculate the cosine of the angle

We know that \(\cos135^{\circ}=-\frac{\sqrt{2}}{2}\approx - 0.707\).

Step3: Substitute values into the Law - of - Cosines formula

\[

$$\begin{align*} d^{2}&=400^{2}+250^{2}-2\times400\times250\times\cos135^{\circ}\\ &=160000 + 62500-200000\times(-0.707)\\ &=160000+62500 + 141400\\ &=363900 \end{align*}$$

\]

Step4: Find the value of \(d\)

\[d=\sqrt{363900}\approx 603.24\approx603.3\text{ miles}\]

Answer:

B. 603.3 miles