Points A and B are separated by a lake. To find the distance between them, a sur

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 = 48.2 degree. He also measures CA as 313 ft and CB as 527 ft. Find the distance between A and B. 167.7 ft A tree on a hillside casts a shadow c = 220 ft down the hill. If the angle of inclination of the hillside is b = 19 degree to the horizontal and the angle of elevation of the sun is a = 49 degree. find the height of the tree. (Round your answer to the nearest foot.) 167.7 ft

Explanation / Answer

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 = 313 feet (given)
BC = 527 feet (given)
angle CAB = 48.2 degrees (given)
angle CAD = 48.2 degrees also because it's the same angle.
sin(CAD) = CD / AC
This becomes:
sin(48.2) = CD / 313
multiply both sides of this equation to get:
313 * sin(48.2) = CD
This makes CD = 233.333987

We use the value of CD to get angle CBD

sin(CBD) = CD / 527

This becomes:
sin(CBD) = 233.333987 / 527 = 0.442758988
This makes angle CBD = 27.301535 degrees.

We can now find AD and BD.

cos(CAD) = AD / AC
This becomes:
cos(48.2) = AD / 313
multiply both sides of this equation by 313 to get:
313 * cos(48.2) = AD
This makes AD = 208.6246631 feet.

cos(CBD) = BD / BC
multiply both sides of this equation by BC to get:
BC * cos(CBD) = BD
This becomes:
527 * cos(27.301535) = BD
This makes BD = 468.29480588 feet

Total length of AB = AD + BD = 208.6246631 + 468.29480588 = 676.9194689 feet.

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