Jackie is standing at the edge of a $150 \mathrm{~m}$ cliff that overlooks a large forest. When she looks over the forest she can see two ranger stations. She measures an angle of depression of $35^{\circ}$ to the first ranger station. She also measures an angle of depression of $20^{\circ}$ to the second ranger station. Lastly, she measures the angle between the two ranger stations to be $40^{\circ}$. How far apart are the two ranger stations to the nearest metre?

Step 1 ：Let's denote the height of the cliff as \(h = 150\) m, the angle of depression to the first ranger station as \(\theta_1 = 35^\circ\), the angle of depression to the second ranger station as \(\theta_2 = 20^\circ\), and the angle between the two ranger stations as \(\theta = 40^\circ\).

Step 2 ：We can convert these angles to radians: \(\theta_1 = 0.6108652381980153\) rad, \(\theta_2 = 0.3490658503988659\) rad, and \(\theta = 0.6981317007977318\) rad.

Step 3 ：Using the tangent of the angles of depression and the height of the cliff, we can calculate the distances to the ranger stations: \(d_1 = h / \tan(\theta_1) = 214.2222010113172\) m and \(d_2 = h / \tan(\theta_2) = 412.12161291819336\) m.

Step 4 ：Finally, we can use the law of cosines to find the distance between the two ranger stations: \(d = \sqrt{d_1^2 + d_2^2 - 2d_1d_2\cos(\theta)} = 284\) m.

Step 5 ：Final Answer: The two ranger stations are approximately \(\boxed{284}\) metres apart.