an airplane leaves city a and flies 124 km due north to city b. it then turns through an angle of 41° eastward and flies 88 km to city c. what angle θ with respect to due north could the pilot have used to fly directly from city a to city c? see the figure below. (the figure is not drawn to scale.) carry your intermediate computations to at least four decimal places. round your answer to the nearest tenth of a degree.

Question

Accepted Answer

# Explanation:
## Step1: Use the Law of Cosines to find the length of $ AC $
The Law of Cosines formula is $ c^{2}=a^{2}+b^{2}-2ab\cos C $. Here, $ a = 124 $, $ b = 88 $, and $ C=180^{\circ}- 41^{\circ}=139^{\circ}$.
$$
\begin{align*}
AC^{2}&=124^{2}+88^{2}-2\times124\times88\times\cos(139^{\circ})\
AC^{2}&=15376 + 7744-21824\times(- 0.7547)\
AC^{2}&=15376+7744 + 16470.3728\
AC^{2}&=39590.3728\
AC&\approx198.97
\end{align*}
$$
## Step2: Use the Law of Sines to find the angle $ \theta $
The Law of Sines formula is $ \frac{\sin\theta}{88}=\frac{\sin(139^{\circ})}{198.97}$.
$$
\begin{align*}
\sin\theta&=\frac{88\times\sin(139^{\circ})}{198.97}\
\sin\theta&=\frac{88\times0.6561}{198.97}\
\sin\theta&=\frac{57.7368}{198.97}\
\sin\theta&\approx0.2902\
\theta&\approx16.9^{\circ}
\end{align*}
$$

# Answer:
$16.9^{\circ}$

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QUESTION IMAGE

Question

Explanation:

Step1: Use the Law of Cosines to find the length of \( AC \)

Step2: Use the Law of Sines to find the angle \( \theta \)

Answer: