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  1. Problems
  2. /From the top of a cliff 60 m high, the angle of depression of a boat is 30°…
HardMathematicsHigh School (9–10)GeneralTrigonometry › Heights and DistancesNumericalword problem

From the top of a cliff 60 m high, the angle of depression of a boat is 30°…

StemPy LabContributed by StemPy Lab

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

From the top of a cliff 606060 m high, the angle of depression of a boat is 30°30°30°. The boat sails straight towards the foot of the cliff. After 121212 seconds, the angle of depression is 45°45°45°. Find the distance moved, the speed, and the time still needed to reach the foot.

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Work it out on paper, then type your number. The answer itself stays hidden.

Need a reminder? The key ideaShowHide

For an object at a fixed height hhh, the horizontal distance at each sighting is hcot⁡θh\cot\thetahcotθ. The distance moved is the difference of the two distances, and speed is the distance moved divided by the time. A width is a difference of two distances along one line.

The worked solution

Have you tried it?

Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

Step 1. The two distances. 60cot⁡30°=603≈103.9260\cot 30° = 60\sqrt{3} \approx 103.9260cot30°=603​≈103.92 m, and 60cot⁡45°=6060\cot 45° = 6060cot45°=60 m.

Step 2. The distance moved. 103.92−60=43.92103.92 - 60 = 43.92103.92−60=43.92 m.

Step 3. The speed. 43.9212=3.66\dfrac{43.92}{12} = 3.661243.92​=3.66 m/s.

Step 4. The time left. 603.66≈16.39\dfrac{60}{3.66} \approx 16.393.6660​≈16.39 s.

Answer: 43.9243.9243.92 m, at 3.663.663.66 m/s, with 16.3916.3916.39 s still to go.

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