Question

two docks are located on an east - west line 2591 ft apart. from dock a, the bearing of a coral reef is 63°22. from dock b, the bearing of the coral reef is 333°22. find the distance from dock a to the coral reef. (round to the nearest integer as needed.)

Explanation:

Step1: Convert angles to decimal degrees

First, convert $63^{\circ}22'$ and $333^{\circ}22'$ to decimal - degrees.
$63^{\circ}22'=63+\frac{22}{60}\approx63.3667^{\circ}$
$333^{\circ}22' = 333+\frac{22}{60}\approx333.3667^{\circ}$
The angle at the coral reef $\theta=360^{\circ}-333.3667^{\circ}+63.3667^{\circ}=90^{\circ}$

Step2: Set up a right - triangle model

Let the distance between the two docks be $AB = 2591$ ft. Let the distance from dock A to the coral reef be $AC$. Let the distance from dock B to the coral reef be $BC$. Since the angle at the coral reef is $90^{\circ}$, we can use trigonometric relations or the Pythagorean theorem.
Let the angle at dock A be $\alpha = 63.3667^{\circ}$.
We know that $\sin\alpha=\frac{BC}{AB}$ and $\cos\alpha=\frac{AC}{AB}$

Step3: Calculate the distance from dock A to the coral reef

Since $\cos\alpha=\frac{AC}{AB}$, then $AC = AB\cos\alpha$
Substitute $AB = 2591$ ft and $\alpha=63.3667^{\circ}$
$AC = 2591\times\cos(63.3667^{\circ})$
$AC = 2591\times0.447$
$AC\approx1158$ ft

Answer:

1158