Question

page 15 of 15 Solving Right Triangles Quick Check 2 of 52 of 5 Items Question Salvador 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?(1 point) Responses 131 feet 131 feet 71 feet 71 feet 48 feet 48 feet 144 feet 144 feet Skip to navigation

Answer 1

To find the horizontal distance from the bridge to the boat, we can use trigonometry. The angle of depression is 40 degrees.

1. First, draw a right triangle where:
- The height of the triangle (opposite side) is the height of the bridge, which is 110 feet.
- The angle of depression from the bridge is 40 degrees.
- The horizontal distance to the boat is the adjacent side we want to find.

2. Use the tangent function, defined as:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Here, \(\theta\) is the angle of depression (40 degrees), the opposite side is 110 feet, and the adjacent side is the horizontal distance we want to find (let's call it \(d\)):
\[
\tan(40^\circ) = \frac{110}{d}
\]

3. Rearranging gives us:
\[
d = \frac{110}{\tan(40^\circ)}
\]

4. Now, calculate \(\tan(40^\circ)\) using a calculator:
\[
\tan(40^\circ) \approx 0.8391
\]

5. Substitute this value back into the equation:
\[
d \approx \frac{110}{0.8391} \approx 130.5
\]

6. Rounding to the nearest foot, the horizontal distance \(d\) is approximately:
\[
d \approx 131 \text{ feet}
\]

So, the boat is approximately **131 feet** from Salvador and Armando.