Oliver is at the top of a lighthouse looking
down at a boat in the water. The angle of
depression from Oliver to the boat is 55
degrees. The boat is 100 feet away from
the light house. How high up is Oliver?
Round your answer to the nearest tenth of
a foot.



Answer :

To determine how high up Oliver is in the lighthouse, here's a step-by-step solution:

1. Understand the Problem:
- Angle of depression from Oliver to the boat: 55 degrees.
- Horizontal distance from the boat to directly below the lighthouse: 100 feet.
- We need to find the height of Oliver above the water.

2. Concepts Involved:
- The angle of depression from the top of the lighthouse to the boat is equal to the angle of elevation from the boat to the top of the lighthouse.
- Trigonometric function tangent (tan) relates the angle of elevation to the height (opposite side) and the horizontal distance (adjacent side).

3. Trigonometric Relationship:
[tex]\[ \tan(\text{angle of elevation}) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
Here,
- [tex]\(\text{angle of elevation} = 55\)[/tex] degrees.
- [tex]\(\text{adjacent} = 100\)[/tex] feet (horizontal distance from the boat to the lighthouse).
- [tex]\(\text{opposite} = \text{height of the lighthouse}\)[/tex].

4. Using the Tangent Function:
[tex]\[ \tan(55^\circ) = \frac{\text{height}}{100 \text{ feet}} \][/tex]

5. Solve for the Height:
[tex]\[ \text{height} = 100 \times \tan(55^\circ) \][/tex]
Use a calculator to determine [tex]\(\tan(55^\circ)\)[/tex]:
[tex]\[ \tan(55^\circ) \approx 1.4281 \][/tex]

6. Calculate the Height:
[tex]\[ \text{height} = 100 \times 1.4281 = 142.81 \text{ feet} \][/tex]

7. Round to the Nearest Tenth of a Foot:
[tex]\[ \text{height} \approx 142.8 \text{ feet} \][/tex]

Therefore, Oliver is approximately 142.8 feet above the water.