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

48 feet
48 feet

71 feet
71 feet

144 feet
144 feet

Answer 1

To determine the horizontal distance from Salvador and Armando to the boat, we can use the concept of right triangles and trigonometry.

In this scenario, we have:

• The height of the bridge (which is the opposite side of the triangle) = 110 feet.
• The angle of depression (which is also the angle of elevation from the boat to the bridge) = 40 degrees.

We need to find the horizontal distance (adjacent side) from the bridge to the boat. We can use the tangent function, which is defined as:

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

Rearranging the equation to solve for the horizontal distance (adjacent) gives us:

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

Substituting the known values:

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

Now we calculate \(\tan(40^\circ)\):

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

Now we can calculate the horizontal distance:

\[ \text{adjacent} = \frac{110}{0.8391} \approx 131.06 \text{ feet} \]

Rounding to the nearest foot, the horizontal distance from Salvador and Armando to the boat is approximately 131 feet.

Therefore, the answer is:

131 feet.