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.

Explanation:

Step1: Set up the trigonometric relationship

We know that in a right - triangle (where the height of the lighthouse is the opposite side and the horizontal distance is the adjacent side with respect to the angle of elevation), the tangent function is given by $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Let $x$ be the horizontal distance. Here, $\theta = 4^{\circ}$ and the opposite side $y = 135$ feet. So, $\tan(4^{\circ})=\frac{135}{x}$.

Step2: Solve for $x$

We can rewrite the equation from Step 1 as $x=\frac{135}{\tan(4^{\circ})}$. We know that $\tan(4^{\circ})\approx0.06992681$. Then $x=\frac{135}{0.06992681}$.

Step3: Calculate the value of $x$

$x=\frac{135}{0.06992681}\approx1930.56$

Answer:

$1930.56$ feet