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)

Question

Responses

144 feet
144 feet

71 feet
71 feet

48 feet
48 feet

131 feet

Answer 1

To determine how far horizontally the boat is from Salvador and Armando, we can use trigonometry. The angle of depression is the angle formed with the horizontal line from their position on the bridge to the line of sight down to the person in the boat.

1. The height of the bridge above the water gives us the opposite side of the triangle: \(h = 110 \text{ feet}\).
2. The angle of depression is \(40^\circ\).
3. We want to find the horizontal distance \(d\) to the boat, which represents the adjacent side of the triangle.

Using the tangent of the angle, which relates the opposite side to the adjacent side, we have:

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

Replacing this with our variables:

\[ \tan(40^\circ) = \frac{110 \text{ feet}}{d} \]

To solve for \(d\), we manipulate the equation:

\[ d = \frac{110 \text{ feet}}{\tan(40^\circ)} \]

Now, we need to calculate \(\tan(40^\circ)\). Using a calculator, we find:

\[ \tan(40^\circ) \approx 0.8391 \]

Now, plug this value into the equation:

\[ d = \frac{110}{0.8391} \approx 131.0 \text{ feet} \]

Rounding to the nearest foot, the horizontal distance to the boat is approximately:

\[ \boxed{131 \text{ feet}} \]