1. **Stating the problem:** We have a circle with center M and radius $r = 6370$ km. There is a sector formed by two radii with a central angle of $142^\circ$. We want to find the length $h$ of the vertical line from the top of the circle that forms a $71^\circ$ angle with the tangent line at the top point. 2. **Understanding the geometry:** The angle between the vertical line $h$ and the tangent line is $71^\circ$. Since the tangent at the top point is perpendicular to the radius, the angle between the radius and the vertical line $h$ is $90^\circ - 71^\circ = 19^\circ$. 3. **Using trigonometry:** In the right triangle formed by the radius, the vertical line $h$, and the line from the center to the point where $h$ meets the circle, we use the cosine of $19^\circ$: $$\cos(19^\circ) = \frac{r}{r + h}$$ Rearranging to solve for $h$: $$r + h = \frac{r}{\cos(19^\circ)}$$ $$h = \frac{r}{\cos(19^\circ)} - r$$ 4. **Calculating $h$:** $$h = 6370 \times \left(\frac{1}{\cos(19^\circ)} - 1\right)$$ Using $\cos(19^\circ) \approx 0.9455$: $$h = 6370 \times \left(\frac{1}{0.9455} - 1\right) = 6370 \times (1.057 - 1) = 6370 \times 0.057 = 362.1$$ 5. **Final answer:** $$h \approx 362.1 \text{ km}$$ This means the vertical line $h$ is approximately 362.1 km long.