Question

question
a boat is heading towards a lighthouse, whose beacon - light is 135 feet above the water. the boats crew measures the angle of elevation to the beacon, 4°. what is the ships horizontal distance from the lighthouse (and the shore)? round your answer to the nearest hundredth of a foot if necessary.
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Explanation:

Step1: Identify the trigonometric relationship

We have a right - triangle situation where the height of the lighthouse (opposite side) is \(h = 135\) feet and the angle of elevation \(\theta=4^{\circ}\). We want to find the horizontal distance \(x\) (adjacent side) from the boat to the lighthouse. We use the tangent function \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).
So, \(\tan\theta=\frac{h}{x}\), which can be rewritten as \(x = \frac{h}{\tan\theta}\).

Step2: Substitute the values

Given \(h = 135\) feet and \(\theta = 4^{\circ}\). We know that \(\tan(4^{\circ})\approx0.0699\) (using a calculator, \(\tan\theta=\tan(4^{\circ})\) in degree mode).
Substitute into the formula \(x=\frac{135}{\tan(4^{\circ})}\).
\(x=\frac{135}{0.0699}\approx1931.33\)

Answer:

\(1931.33\)