A man standing on a lighthouse of height 162 ft sights tv
boats directly in front of him. One is at an angle of
depression of , and the other is at an angle of
depression of . Identify the distance between the two
boats rounded to the nearest foot.

Ask by Mejia Collins. in the United States

Mar 26,2025

Question

A man standing on a lighthouse of height 162 ft sights tv
boats directly in front of him. One is at an angle of
depression of , and the other is at an angle of
depression of . Identify the distance between the two
boats rounded to the nearest foot.

Ask by Mejia Collins. in the United States

Mar 26,2025

UpStudy AI · Accepted Answer

Let the distances from the base of the lighthouse to the two boats be $$ d_{50} $$ (for the boat with a $$50^\circ$$ angle of depression) and $$ d_{45} $$ (for the boat with a $$45^\circ$$ angle of depression). The height of the lighthouse is $$162$$ ft.

Since the angle of depression equals the angle of elevation from the boat, we use the tangent function in a right triangle. For a given angle $$ \theta $$,

$$
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{162}{d}.
$$

1. For the $$50^\circ$$ angle:
   $$
   \tan(50^\circ) = \frac{162}{d_{50}} \quad \Longrightarrow \quad d_{50} = \frac{162}{\tan(50^\circ)}.
   $$

2. For the $$45^\circ$$ angle:
   $$
   \tan(45^\circ) = \frac{162}{d_{45}} \quad \Longrightarrow \quad d_{45} = \frac{162}{\tan(45^\circ)} = \frac{162}{1} = 162 \text{ ft}.
   $$

Next, we compute $$ d_{50} $$. Using an approximation,
   $$
   \tan(50^\circ) \approx 1.1918,
   $$
   so,
   $$
   d_{50} \approx \frac{162}{1.1918} \approx 135.9 \text{ ft}.
   $$

The distance between the two boats is the difference between $$ d_{45} $$ and $$ d_{50} $$:
   $$
   \text{Distance} \approx 162 - 135.9 \approx 26.1 \text{ ft}.
   $$

Rounded to the nearest foot, the distance between the two boats is $$26$$ ft.
The distance between the two boats is approximately 26 feet.

A man standing on a lighthouse of height 162 ft sights tv
boats directly in front of him. One is at an angle of
depression of , and the other is at an angle of
depression of . Identify the distance between the two
boats rounded to the nearest foot.

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A man standing on a lighthouse of height 162 ft sights tv boats directly in front of him. One is at an angle of depression of , and the other is at an angle of depression of . Identify the distance between the two boats rounded to the nearest foot.

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Latest Trigonometry Questions

A man standing on a lighthouse of height 162 ft sights tv
boats directly in front of him. One is at an angle of
depression of , and the other is at an angle of
depression of . Identify the distance between the two
boats rounded to the nearest foot.