A simple measuring device is used at points X and Y on the same horizontal level to measure the angles of elevation of the peak P of a certain mountain. If ...

Let the foot of the perpendicular from the peak \(P\) to the horizontal level of \(X\) and \(Y\) be \(N\), with \(Y\) nearer the mountain. Let \(|YN| = d\) and let \(h\) be the height of \(P\) above the level \(XY\).

From \(Y\) (angle of elevation \(35^\circ\)):

\[\tan 35^\circ = \frac{h}{d} \;\Rightarrow\; h = 0.7d\]

From \(X\) (angle of elevation \(15^\circ\), and \(|XN| = d + 4000\)):

\[\tan 15^\circ = \frac{h}{d + 4000} \;\Rightarrow\; h = 0.3(d + 4000)\]

Equating:

\[0.7d = 0.3d + 1200 \;\Rightarrow\; 0.4d = 1200 \;\Rightarrow\; d = 3000\text{ m}\]

\[h = 0.7(3000) = 2100\text{ m}\]

This \(h\) is the height of the peak above the level of \(X\) and \(Y\). Since \(X\) (and \(Y\)) are \(5200\) m above sea level, the height of the mountain above sea level is:

\[5200 + 2100 = 7300\text{ m}\]