The settlements $M, N, P$ are connected by straight roads. The road from $M$ to $N$ is in a southwesterly direction, is $6 \mathrm{~km}$ long, and has an incline of $14^{\circ}$. From $N$ to $P$, we travel eastward on a $7 \mathrm{~km}$ long road with an incline of $8^{\circ}$. What is the length of the $M P$ road, and what is its incline?
The lengths of the projections of the first two segments on the horizontal plane passing through point $M$ are $M N^{\prime}=6 \cos 14^{\circ} \approx 5.822 \mathrm{~km}$ and $N^{\prime} P^{\prime}=7 \cos 8^{\circ} \approx 6.932 \mathrm{~km}$, with the angle between them being $45^{\circ}$. Thus, using the cosine rule, we can calculate the length of the projection of $M P$: $M P^{\prime} \approx 4.988 \mathrm{~km}$. The height of settlement $N$ relative to $M$ is $6 \sin 14^{\circ} \approx 1.452 \mathrm{~km}$, and the height of settlement $P$ relative to $N$ is $7 \sin 8^{\circ} \approx 0.974 \mathrm{~km}$, so the height difference between $M$ and $P$ is $P^{\prime} P \approx 2.426 \mathrm{~km}$. Therefore, in the right-angled triangle $M P P^{\prime}$, the length of the path $M P$ is $M P \approx 5.55 \mathrm{~km}$, and the elevation angle $\delta$ is given by $\tan \delta = P^{\prime} P / M P^{\prime}$, so $\delta \approx 25.9^{\circ}$.

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Ferenc Naszályi (Budapest, Kölcsey F. g. III. o. t.)