Question 418669: A tree on a hillside casts a shadow c ft down the hill. If the angle of inclination of the hillside is b to the horizontal and the angle of elevation of the sun is a, find the height of the tree. Assume a = 57°, b = 27° c = 240 ft. (Round your answer to the nearest whole number.)
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! A tree on a hillside casts a shadow c ft down the hill. If the angle of inclination of the hillside is b to the horizontal and the angle of elevation of the sun is a, find the height of the tree. Assume a = 57°, b = 27° c = 240 ft. (Round your answer to the nearest whole number.)
..
Draw a hillside that is inclined 27 deg to the horizontal(from right to left).
Someplace on the surface of the hill draw a vertical line (to the horizontal)which represents the height of the tree. From the top of this vertical line we will call point A, draw a line down to the surface of the hill at point B. This is a 33 deg. angle the sun makes with the surface of the hillside. The bottom of the vertical line to point B along the surface is given to be 240 ft.which is the shadow of the tree. Extend the bottom (point D) of the vertical line into the hillside,then at a point we will call C, turn at a right angle to extend this line to point B. We now have two triangles to work with,ABC, a right triangle with angle at B=57 deg. and BCD, a right triangle with angle at B=27 deg., both sharing leg BC.
..
Calculations:
Height of tree=AC-DC=AD
DC/240=Sin 27 deg
DC=240*Sin 27 deg=108.96 ft
AC/CB=tan 57 deg
AC=CB*tan 57 deg
CB/240=Cos 27 deg
CB=240*Cos 27 deg=213.84 ft
AC=213.84*tan 57 deg=329.28 ft
AD=AC-DC=329.28-108.96=220 ft
ans:
Height of tree=220 ft
..
An alternative solution:
Find the angles at A, B & D
At B, it is 57-27=30 deg.
At A, it is 33 deg.
sin 30/AD=sin 33/240
AD=240*SinB/sinA=240*sin30/sin33=220 ft
|
|
|
