SOLUTION: 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 angleCAB = 48.6°. He also measures CA as 316 ft and CB as

Question 233387: 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 angleCAB = 48.6°. He also measures CA as 316 ft and CB as 527 ft. Find the distance between A and B. Round your answer to the nearest foot. (Note: angleABC is an acute angle.)
Answer by Theo(2978) (Show Source):
You can put this solution on YOUR website!
Drop a line perpendicular to line AC and intersecting line AC at point D.

That forms 2 right triangles.

First is triangle ACD and second is triangle BCD.

You have:

AC = 316 feet (given)
BC = 527 feet (given)
angle CAB = 48.6 degrees (given)
angle CAD = 58.6 degrees also because it's the same angle.
sin(CAD) = CD / AC
This becomes:
sin(48.6) = CD / 316
multiply both sides of this equation to get:
316 * sin(48.6) = CD
This makes CD = 237.035098

We use the value of CD to get angle CBD

sin(CBD) = CD / 527
This becomes:
sin(CBD) = 237.035098/527 = .44978197
This makes angle CBD = 26.72969621 degrees.

We can now find AD and BD.

cos(CAD) = AD / AC
This becomes:
cos(48.6) = AD / 316
multiply both sides of this equation by 316 to get:
316 * cos(48.6) = AD
This makes AD = 208.9745494 feet.

cos(CBD) = BD / BC
multiply both sides of this equation by BC to get:
BC * cos(CBD) = BD
This becomes:
527 * cos(26.72969621) = BD
This makes BD = 470.6859304 feet

Total length of AB = AD + BD = 208.9745494 + 470.6859304 = 679.6584798 feet.