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.
