Points A and B are separated by a lake To find the distance

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 angle CAB = 48.9 degree. He also measures CA as 312 ft and CB as 527 ft. Find the distance between A and B. () ft

Solution

Drop a line perpendicular to line AC and intersecting line AC at point D

AC = 312 feet (given)
BC = 527 feet (given)
angle CAB = 48.9 degrees (given)
angle CAD = 48.9 degrees also because it\'s the same angle.
sin(CAD) = CD / AC
This becomes:
sin(48.9) = CD / 312
multiply both sides of this equation to get:
312 * sin(48.9) = CD
This makes CD = 235.11177

We use the value of CD to get angle CBD

sin(CBD) = CD / 527

This becomes:
sin(CBD) = 235.11177/527 =0.4461324068
This makes angle CBD = 26.495812612 degrees.

We can now find AD and BD.

cos(CAD) = AD / AC
This becomes:
cos(48.6) = AD / 312
multiply both sides of this equation by 316 to get:
312 * cos(48.6) = AD
This makes AD = 205.1010766 feet.

Then it becomes-------------

cos(CBD) = BD / BC
multiply both sides of this equation by BC to get:
BC * cos(CBD) = BD
This becomes:
527 * cos(26.495812612) = BD
This makes BD = 471.64759266 feet

Total length of AB = AD + BD = 205.1010766 + 471.64759266 = 676.748669264 feet.