The problem asks us to find the height of a cliff on the opposite side of a river. According to the diagram, a surveyor is measuring two angles from two different positions and the distance between the two points of observation.

Let's break down the given information from the image:

1. The angle from the first position (closer to the river) is $35^\circ$.
2. The angle from the second position is $22^\circ$.
3. The distance between the two observation points is $400.22 \, \text{m}$.

The goal is to find the height of the cliff, denoted as $h$, which forms a right triangle with the base of the river.

Steps:

1. Label the points on the diagram:

   • Let point $A$ be the first position (where the $35^\circ$ angle is measured),
   • Let point $B$ be the second position (where the $22^\circ$ angle is measured),
   • Let point $C$ be the point directly beneath the cliff on the other side of the river.
   • Let $D$ be the top of the cliff.

2. We need to apply the tangent function for both angles to form two equations: $\tan(35^\circ) = \frac{h}{d}$ where $d$ is the distance from point $A$ to the point directly beneath the cliff on the opposite side.

   $\tan(22^\circ) = \frac{h}{d + 400.22}$ This equation comes from the second observation point, accounting for the extra distance of $400.22 \, \text{m}$.

We now have two equations with two unknowns (height $h$ and distance $d$):

$1. \ h = d \cdot \tan(35^\circ)$ $2. \ h = (d + 400.22) \cdot \tan(22^\circ)$

By solving this system of equations, we can find both $h$ and $d$. Let's compute this now.The calculations show the following results:

• The distance from the first observation point to the base of the cliff ($d$) is approximately 545.95 meters.
• The height of the cliff ($h$) is approximately 382.28 meters.

Let me know if you need further details or have any questions.

Here are 5 related questions for deeper understanding:

1. How would the height change if the distance between observation points increased?
2. What would happen to the height calculation if only one angle measurement was available?
3. How can the tangent function be used to solve real-world problems like this one?
4. How does the accuracy of angle measurement affect the height calculation?
5. Could this problem be solved using the sine or cosine function? How?

Tip: When solving trigonometric problems involving distances and angles, ensure that all angle measurements are accurate and that you account for any extra distances between points of observation.