Question

a boat is heading towards a lighthouse, whose beacon - light is 126 feet above the water. the boat’s crew measures the angle of elevation to the beacon, 15°. what is the ship’s horizontal distance from the lighthouse (and the shore)? round your answer to the nearest tenth of a foot if necessary.

Explanation:

Step1: Set up tangent - ratio

We know that in a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, the height of the lighthouse (opposite side) is 126 feet and the angle of elevation $\theta = 15^{\circ}$, and the horizontal distance $x$ is the adjacent side. So, $\tan(15^{\circ})=\frac{126}{x}$.

Step2: Solve for $x$

We can rewrite the equation as $x=\frac{126}{\tan(15^{\circ})}$. Since $\tan(15^{\circ}) = 2-\sqrt{3}\approx0.2679$, then $x=\frac{126}{0.2679}$.

Step3: Calculate the value of $x$

$x=\frac{126}{0.2679}\approx470.3$.

Answer:

470.3 feet