Question

Points A and B are separated by a lake. To find the distance between them, a surveyor locates a point C on land such that CAB = 48.6°. He also measures CA as 310 ft and CB as 527 ft. Find the distance between A and B. Round your answer to the nearest foot. (Note: ABC is an acute angle.)

Answer 1

Use the Law of Cosine
$c^2=a^2+b^2-2ab\cos{\gamma}$

$527^2=310^2+AB^2-2\times310\times AB\times\cos{48.6^o} \\AB^2-410\times AB-181,629=0 \\ \\AB= \frac{410+ \sqrt{410^2-4(-181629)} }{2} =678$