To find how far away the ship is from the base of the lighthouse using the sine ratio, you can set up the problem as follows:
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Identify the triangle: The height of the lighthouse represents the opposite side, the distance from the base of the lighthouse to the ship represents the adjacent side, and the angle of depression from the top of the lighthouse to the ship is 17°.
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Using the sine ratio: The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
In this case, our opposite side is the height of the lighthouse (40 feet) and we'll let \(d\) be the distance from the base of the lighthouse to the ship, while the hypotenuse can be identified as the line of sight from the top of the lighthouse to the ship.
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Finding the distance: The angle of depression is equal to the angle of elevation from the ship to the top of the lighthouse. Thus, we can use: \[ \tan(17^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{40}{d} \]
Rearranging this, we have: \[ d = \frac{40}{\tan(17^\circ)} \]
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Calculating \(d\):
- First, calculate \(\tan(17^\circ)\) using a calculator: \[ \tan(17^\circ) \approx 0.3057 \]
- Now, compute \(d\): \[ d \approx \frac{40}{0.3057} \approx 130.8 \text{ feet} \]
Thus, rounding to the nearest tenth, the distance from the base of the lighthouse to the ship is approximately 130.8 feet.
So the correct response is 130.8 feet.
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