A hiker

A hiker plans to hike up one side of a mountain and down the other side of points a mountain, each side of the mountain formed by a straight line. The angle of elevation at the starting point is 42.4 degrees, and the angle of elevation at the end is 48.3 degrees. The horizontal distance between the start and end points is 19.06 miles. To the nearest tenth, what is the total distance the hiker traveled?

Final Answer:

d =  27.1 mi

Step-by-step explanation:

$$\begin{gathered} \alpha =42.4{}^{\circ } \\ \beta =48.3{}^{\circ } \\ x=19.06\text{mi} \\ x=x_1+x_2 \\ \tan \alpha =h:x_1 \\ \tan \beta =h:x_2 \\ {x}_1=h/\tan \alpha \\ {x}_2=h/\tan \beta \\ x=h/\tan \alpha +h/\tan \beta \\ x=h\cdot (1/\tan \alpha +1/\tan \beta ) \\ h=x/(1/\tan \alpha +1/\tan \beta )=x/(1/\tan 42.4°+1/\tan 48.3°)=19.06/(1/\tan 42.4°+1/\tan 48.3°)=19.06/(1/0.913125+1/1.122375)=9.59666\text{mi} \\ \sin \alpha =h:d_1 \\ \sin \beta =h:d_2 \\ {d}_1=h/\sin \alpha =h/\sin 42.4°=9.5967/\sin 42.4°=9.5967/0.674302=14.23199\text{mi} \\ {d}_2=h/\sin \beta =h/\sin 48.3°=9.5967/\sin 48.3°=9.5967/0.746638=12.85316\text{mi} \\ d=d_1+d_2=14.232+12.8532=27.1\text{mi} \end{gathered}$$

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