Question

the top of a lighthouse is 100 m above sea level. the angle of elevation from the deck of the sailboat to the top of the lighthouse is 28°. calculate the distance between the sailboat and the lighthouse.

Explanation:

Step1: Identify the trigonometric relation

We have a right - triangle where the height of the lighthouse (opposite side) is 100 m and the angle of elevation is 28°. We use the tangent function since $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Let the distance between the sailboat and the lighthouse (adjacent side) be $x$. So, $\tan(28^{\circ})=\frac{100}{x}$.

Step2: Solve for $x$

We can rewrite the equation as $x = \frac{100}{\tan(28^{\circ})}$. Since $\tan(28^{\circ})\approx0.5317$, then $x=\frac{100}{0.5317}\approx188.08$ m.

Answer:

Approximately 188.08 m