To solve for the distance from the lighthouse operator to the ship, we will use the sine ratio from trigonometry. The sine ratio relates the opposite side of a right triangle to the hypotenuse. In this scenario:
- The height of the lighthouse is the opposite side (40 feet).
- The distance from the lighthouse operator to the ship is the hypotenuse (denote this as \( x \)).
- The angle of elevation (the angle from the ship to the top of the lighthouse) is 73°.
The sine function is defined as: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
For our situation, we can write the equation as follows: \[ \sin(73°) = \frac{40}{x} \]
To solve for \( x \), we can rearrange this equation: \[ x = \frac{40}{\sin(73°)} \]
Next, we will calculate \( \sin(73°) \). Typically, you would use a calculator to find this value.
Using a calculator: \[ \sin(73°) \approx 0.9563 \]
Now we can substitute this value back into our equation to find \( x \): \[ x = \frac{40}{0.9563} \approx 41.85 \]
Therefore, the distance from the lighthouse operator to the ship is approximately 41.85 feet.
Final answer: The ship is approximately 41.85 feet from the lighthouse operator at the top of the lighthouse.
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