Question

two ships leave a harbor together traveling on courses that have an angle of 121° between them. if they each travel 535 miles, how far apart are they (to the nearest mile)?

a. 527 mi
b. 931 mi
c. 40 mi
d. 1862 mi

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(c^{2}=a^{2}+b^{2}-2ab\cos C\). Here, \(a = 535\), \(b = 535\), and \(C=121^{\circ}\).
Substitute the values into the formula: \(c^{2}=535^{2}+535^{2}-2\times535\times535\times\cos(121^{\circ})\).
First, calculate \(535^{2}=286225\). Then, \(2\times535\times535 = 2\times286225=572450\).
We know that \(\cos(121^{\circ})\approx - 0.5150\).

Step2: Calculate \(c^{2}\)

\(c^{2}=286225 + 286225-572450\times(-0.5150)\)
\(c^{2}=572450+572450\times0.5150\)
\(c^{2}=572450(1 + 0.5150)\)
\(c^{2}=572450\times1.515\)
\(c^{2}=867261.75\)

Step3: Find \(c\)

Take the square - root of \(c^{2}\): \(c=\sqrt{867261.75}\approx931\)

Answer:

B. 931 mi