Question

ex 3: the bearing from a to c is s 52° e. the bearing from a to b is n 84° e. the bearing from b to c is s 38° w. a plane flying at 250 mph takes 2.4 hours to go from a to b. find the distance from a to c.

Explanation:

Step1: Calculate distance from A to B

We know speed $v = 250$ mph and time $t=2.4$ hours. Using the formula $d = vt$, we have $d_{AB}=250\times2.4$.
$d_{AB}=600$ miles

Step2: Find angles in the triangle

The angle $\angle BAC=84^{\circ}+(90 - 52)^{\circ}=122^{\circ}$. The angle $\angle ABC = 84^{\circ}-38^{\circ}=46^{\circ}$. Then the angle $\angle ACB=180^{\circ}-122^{\circ}-46^{\circ}=12^{\circ}$.

Step3: Use the Law of Sines

By the Law of Sines, $\frac{d_{AB}}{\sin\angle ACB}=\frac{d_{AC}}{\sin\angle ABC}$. Substitute $d_{AB} = 600$, $\angle ACB = 12^{\circ}$, and $\angle ABC=46^{\circ}$. So $d_{AC}=\frac{600\times\sin46^{\circ}}{\sin12^{\circ}}$.
$\sin46^{\circ}\approx0.7193$, $\sin12^{\circ}\approx0.2079$. Then $d_{AC}=\frac{600\times0.7193}{0.2079}\approx2073.6$.

Answer:

Approximately 2074 miles