Angles of elevation and depression

Question

A man stands on a cliff 80m high. He watches a boat from the cliff and notices that the angle of depression from the cliff to the boat changes from 45° to 30°. What distance did the boat cover while the man was watching it?

Answer 1

Let's denote the distance covered by the boat as \(x\).

From the information provided, we can see that we have a right triangle formed by the cliff, the boat, and the line of sight from the man. The angle of elevation from the boat to the top of the cliff is \(45°\), and the angle of depression from the cliff to the boat is \(30°\).

Using trigonometry, we can set up the following equation:

\[
\begin{aligned}
\tan(45°) & = \frac{80}{x - 80} \\
\tan(30°) & = \frac{80}{x}
\end{aligned}
\]

Now, we can solve these equations to find the value of \(x\):

\[
\begin{aligned}
x - 80 & = 80 \cdot \tan(45°) \\
x - 80 & = 80 \\
x & = 160
\end{aligned}
\]

Therefore, the boat covered a distance of 160 meters while the man was watching it.