Kiara's yacht is anchored 90 feet offshore from the base of a lighthouse. The angle of elevation from the boat to the top of the lighthouse is 26 degrees. The distance between the yacht and the top of the lighthouse is about 100 feet.Find the height of the lighthouse.

To find the height of the lighthouse, we can use trigonometric relationships. In this case, we have a right triangle formed by the height of the lighthouse (\( h \)), the distance from the yacht to the base of the lighthouse (90 feet), and the hypotenuse (the distance from the yacht to the top of the lighthouse, 100 feet).

### Step 1: Identify the relationship

Given:
- Distance from the yacht to the base of the lighthouse = 90 feet
- Angle of elevation = 26 degrees
- Hypotenuse (distance from the yacht to the top of the lighthouse) = 100 feet

We need to find the height \( h \) of the lighthouse.

### Step 2: Use the sine function

The sine function relates the angle of elevation to the opposite side (height of the lighthouse) and the hypotenuse:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{100}
\]

### Step 3: Solve for \( h \)
\[
h = 100 \times \sin(26^\circ)
\]

Let's compute the value of \( h \).The height of the lighthouse is approximately **43.84 feet**.

Would you like further details on the solution, or do you have any additional questions?

Here are some related questions you might find interesting:

1. What would the height of the lighthouse be if the angle of elevation was 30 degrees?
2. How would the height change if the yacht were 120 feet offshore?
3. If the distance between the yacht and the top of the lighthouse were 150 feet, what would the height be with the same angle of elevation?
4. Can you calculate the angle of elevation if the height of the lighthouse is 50 feet and the distance from the yacht to the base is 90 feet?
5. How far offshore would the yacht be if the angle of elevation was 45 degrees, given the same lighthouse height?
6. What is the distance from the yacht to the top of the lighthouse if the height is 60 feet and the angle of elevation is 20 degrees?
7. If the angle of elevation were to increase to 35 degrees, how would that affect the height of the lighthouse?
8. How does the angle of elevation affect the calculation of height in such problems?

**Tip:** When dealing with right triangles, it's crucial to identify which trigonometric function to use based on the given information. Sine, cosine, and tangent are key to solving these types of problems.

Learn how to find the height of a lighthouse using trigonometric functions. Given an angle of elevation of 26 degrees, distance from yacht to lighthouse base of 90 feet, and distance to lighthouse top of 100 feet, calculate the lighthouse height step-by-step.

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