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

131 feet
131 feet

144 feet
144 feet

48 feet
48 feet

71 feet
71 feet

Answer 1

To find the horizontal distance from Salvador and Armando to the boat, we can use the tangent function, which relates the angle of depression to the opposite side (the height of the bridge) and the adjacent side (the horizontal distance to the boat).

Given:

• Height of the bridge (opposite side) = 110 feet
• Angle of depression = 40 degrees

Using the tangent function: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]

Substituting in the known values: \[ \tan(40^\circ) = \frac{110}{d} \] where \(d\) is the horizontal distance to the boat.

Rearranging to solve for \(d\): \[ d = \frac{110}{\tan(40^\circ)} \]

Now we can calculate \(\tan(40^\circ)\). Using a calculator: \[ \tan(40^\circ) \approx 0.8391 \]

Now, substituting this value in: \[ d \approx \frac{110}{0.8391} \approx 131.0 \text{ feet} \]

Rounding to the nearest foot, we find the horizontal distance is approximately: \[ \text{Horizontal distance} \approx 131 \text{ feet} \]

Thus, the answer is: 131 feet.