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. the distance from dock a to the coral reef ft. (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 dock A, $\angle A = 63.3667^{\circ}$. The angle at dock B, $\angle B=360 - 333.3667=26.6333^{\circ}$. Then the angle at the coral reef, $\angle C=180-(63.3667 + 26.6333)=90^{\circ}$.

Step2: Use the sine - rule

Let the distance between dock A and dock B be $c = 2591$ ft, the distance between dock A and the coral reef be $b$.
By the sine - rule $\frac{b}{\sin B}=\frac{c}{\sin C}$.
Since $\sin C=\sin90^{\circ}=1$ and $\sin B=\sin(26.6333^{\circ})\approx0.448$, and $c = 2591$ ft.
We have $b=\frac{c\sin B}{\sin C}=\frac{2591\times\sin(26.6333^{\circ})}{\sin90^{\circ}}$.
$b = 2591\times0.448\approx1160$ ft.

Answer:

1160