To find how far away the ship is from the top of the lighthouse using the angle of depression and the height of the lighthouse, you can use the sine ratio. Here's how to approach this problem:
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First, let's understand the given values:
- Height of the lighthouse (opposite side of the triangle) = 40 feet
- Angle of depression = 17°
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The angle of depression from the top of the lighthouse to the ship is equal to the angle of elevation from the ship to the top of the lighthouse. Therefore, the angle at the ship is also 17°.
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We will use the tangent function since we need the distance to the ship (the adjacent side) and we know the height of the lighthouse (the opposite side). The tangent function uses the opposite and adjacent sides.
The formula is: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
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Substituting in the known values: \[ \tan(17°) = \frac{40}{d} \] where \(d\) is the distance from the base of the lighthouse to the ship.
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Rearranging gives: \[ d = \frac{40}{\tan(17°)} \]
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Now we will calculate the tangent of 17° and then the value of \(d\): \[ d \approx \frac{40}{0.3057} \approx 130.8 \text{ feet} \]
So, the distance from the ship to the base of the lighthouse is approximately 130.8 feet.
The correct response is: 130.8 feet.
2 answers