#### Brotherbobby

##### New member

- Joined
- Jul 10, 2019

- Messages
- 1

**Statement of the Problem :**A satellite located at a height \(\displaystyle h\) from the base B sends radio signals to your place P. These signals give us the distance \(\displaystyle s\) to your place up to a certain measurement error \(\displaystyle \delta s\). Your place P is located at a distance \(\displaystyle L\) along the earth, as shown in the diagram above. Use the methods of differential calculus to calculate \(\displaystyle L\) to a first approximation.

**Solution :**I Imagine a simplified triangular image of the three points as shown above. The distance BP' = \(\displaystyle L_0 = \sqrt{s_0^2 - h^2}\). The original distance \(\displaystyle L = L_0 +\delta L \approx L_0 + \left( \frac{dL}{ds}\right)_{s_0}\delta s \).

From the triangle above, using the methods of trigonometry and calculus, we get \(\displaystyle \frac{dL}{ds} = \frac{s}{L}\) and hence \(\displaystyle \left( \frac{dL}{ds}\right )_{s_0} = \frac{s_0}{L_0} \text{, hence the original distance}\; L = L_0 + \frac{s_0}{L_0} \delta s \Rightarrow \boxed{L = L_0\left( 1+\frac{s_0}{L_0^2}\delta s \right)}\).

The satellite gives the error in the measurement of \(\displaystyle s\) which is \(\displaystyle \delta s\) and so the distance

*L*can be found.

Is my working correct?