To find the horizontal distance from Salvador and Armando to the boat, we can use the trigonometric relationship involving the angle of depression and the height of the bridge.
In this scenario, the height of the bridge (opposite side) is 110 feet, and the angle of depression is 40 degrees. Since the angle of depression and the angle of elevation from the boat to Salvador and Armando are the same, we can consider the angle of elevation from the boat to the top of the bridge as 40 degrees.
We can use the tangent function, which is defined as:
\[ \tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \]
In this case:
- Opposite = 110 feet (height of the bridge)
- Adjacent = horizontal distance (d) we need to find
- Angle = 40 degrees
Setting up the equation, we have:
\[ \tan(40^\circ) = \frac{110}{d} \]
Now we can solve for \(d\):
\[ d = \frac{110}{\tan(40^\circ)} \]
Now we need to compute \(\tan(40^\circ)\). Using a calculator:
\[ \tan(40^\circ) \approx 0.8391 \]
Now substituting this value back into the equation:
\[ d = \frac{110}{0.8391} \approx 130.4 \text{ feet} \]
Rounding to the nearest foot gives us approximately 130 feet.
Among the response choices, the closest answer is:
131 feet.
1 answer