Question

a ferry transports passengers to three different ports. from port a, the ferry travels east 3.5 miles to port b. the ship then turns 75° south and travels for 6 miles to reach port c. what is the distance d from port c to port a? approximate to the nearest tenth of a mile

Explanation:

1. First, identify the information given:

• In triangle \(ABC\), \(AB = 3.5\) miles, \(BC = 6\) miles, and \(\angle ABC=180 - 75=105^{\circ}\).

1. Then, use the Law - of - Cosines:

• The Law of Cosines states that for a triangle with sides \(a\), \(b\), \(c\) and the angle \(\theta\) opposite to side \(c\), \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\).
• In triangle \(ABC\), if we want to find the length of \(AC = d\), where \(a = 3.5\), \(b = 6\), and \(\theta = 105^{\circ}\), and \(\cos105^{\circ}=\cos(60^{\circ}+45^{\circ})=\cos60^{\circ}\cos45^{\circ}-\sin60^{\circ}\sin45^{\circ}=\frac{1}{2}\times\frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{2}\times\frac{\sqrt{2}}{2}=\frac{\sqrt{2}-\sqrt{6}}{4}\approx - 0.259\).
• According to the Law of Cosines, \(d^{2}=3.5^{2}+6^{2}-2\times3.5\times6\times\cos105^{\circ}\).
• Calculate each part:
• \(3.5^{2}=12.25\), \(6^{2}=36\).
• \(2\times3.5\times6 = 42\).
• \(d^{2}=12.25 + 36-42\times(-0.259)\).
• \(d^{2}=12.25 + 36+10.878\).
• \(d^{2}=59.128\).
• Take the square - root of both sides:
• \(d=\sqrt{59.128}\approx7.7\) miles.

Answer:

7.7