Question

Slavador and Armando use a land surveying tool, a clinometer, to determine the angle of depression from their position on a bridge to a person in a boat on the water. They know the bridge is 110 feet above the water and the angle of depression is 40 degrees. How far (to the nearest foot) horizontally is the boat from Salvador and Armando?

Answer 1

To find the horizontal distance from Salvador and Armando to the boat, we can use the relationship between the angle of depression, the height of the bridge, and the horizontal distance.

1. **Understanding the problem**:
- The angle of depression from the bridge to the boat is given as 40 degrees.
- The height of the bridge above the water is 110 feet.

2. **Identifying the right triangle**:
- The height (opposite side) is 110 feet (the height of the bridge).
- The angle of depression corresponds to the angle of elevation from the boat to the bridge, which is 40 degrees.
- The horizontal distance to the boat is the adjacent side of the triangle we need to find.

3. **Using trigonometric functions**:
- We can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side.

\[
\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}
\]

In our case, we have:
\[
\tan(40^\circ) = \frac{110 \text{ ft}}{d}
\]
where \( d \) is the horizontal distance from the bridge to the boat.

4. **Solving for \( d \)**:
- Rearranging the equation, we have:
\[
d = \frac{110 \text{ ft}}{\tan(40^\circ)}
\]

5. **Calculating \( d \)**:
- First, we need to calculate \( \tan(40^\circ) \).
- Using a calculator, we find:
\[
\tan(40^\circ) \approx 0.8391
\]

Now substitute this value back into the equation:
\[
d \approx \frac{110}{0.8391} \approx 131.01 \text{ ft}
\]

6. **Rounding to the nearest foot**:
- Therefore, the horizontal distance \( d \) is approximately \( 131 \) feet.

Thus, the horizontal distance from Salvador and Armando to the boat is **131 feet**.