To find the horizontal distance from Salvador and Armando to the boat, we can use the relationship between the angle of depression, the height of the bridge, and the horizontal distance.
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Understanding the problem:
- The angle of depression from the bridge to the boat is given as 40 degrees.
- The height of the bridge above the water is 110 feet.
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Identifying the right triangle:
- The height (opposite side) is 110 feet (the height of the bridge).
- The angle of depression corresponds to the angle of elevation from the boat to the bridge, which is 40 degrees.
- The horizontal distance to the boat is the adjacent side of the triangle we need to find.
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Using trigonometric functions:
- We can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side.
\[ \tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \]
In our case, we have: \[ \tan(40^\circ) = \frac{110 \text{ ft}}{d} \] where \( d \) is the horizontal distance from the bridge to the boat.
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Solving for \( d \):
- Rearranging the equation, we have: \[ d = \frac{110 \text{ ft}}{\tan(40^\circ)} \]
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Calculating \( d \):
- First, we need to calculate \( \tan(40^\circ) \).
- Using a calculator, we find: \[ \tan(40^\circ) \approx 0.8391 \]
Now substitute this value back into the equation: \[ d \approx \frac{110}{0.8391} \approx 131.01 \text{ ft} \]
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Rounding to the nearest foot:
- Therefore, the horizontal distance \( d \) is approximately \( 131 \) feet.
Thus, the horizontal distance from Salvador and Armando to the boat is 131 feet.
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