A boat is sighted from a 50-meter observation tower on the shoreline at an angle of depression of 4 degrees moving directly towards the shore at a constant speed. Five minutes later the angle of depression of the boat is 12 degrees. What is the speed of the boat in kilometers per hour?

5.76 km/h

Step-by-step explanation:

The mnemonic SOH CAH TOA reminds you of the relationship between angles and sides of a right triangle. Here, we are given the side opposite the angle (angle of depression), and we want to find the adjacent side (distance from shore).

Tan = Opposite/Adjacent

tan (4°) = (height of tower) / (distance from shore)

tan (4°) = (50 m) / (distance from shore)

Then the distance from shore is ...

distance from shore = (50 m) / tan (4°) ≈ 715.03 m

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At the second sighting, the distance from shore is ...

distance from shore = (50 m) / tan (12°) ≈ 235.23 m

So, the distance traveled in 1/12 hour is ...

715.03 m - 235.23 m = 479.80 m

and the speed in km per hour is ...

speed = 0.4798 km / (1/12 h) = 5.7576 km/h

The speed of the boat is about 5.76 km per hour.