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

71 feet
71 feet

48 feet
48 feet

144 feet
144 feet

131 feet

Answer 1

To determine the horizontal distance from Salvador and Armando to the boat, we can use trigonometry. Specifically, 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

In this case, the formula we will use is:

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

Rearranging it gives us:

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

Plugging in the values:

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

Using a calculator to find $\tan ({40}^{\circ })$:

$\tan ({40}^{\circ })\approx 0.8391$

Now, substitute this value into the equation:

$$\text{adjacent}\approx \frac{110}{0.8391}\approx 131.0\text{ feet}$$

Rounding to the nearest foot, the horizontal distance is approximately 131 feet.

So the correct response is: 131 feet