SOLUTION: A tree on a hillside casts a shadow 45 ft up the hill. If the angle of inclination of the hillside is 12 degrees to the horizontal and the angle of elevation of the sun is 32 degre

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Question 1187688: A tree on a hillside casts a shadow 45 ft up the hill. If the angle of inclination of the hillside is 12 degrees to the horizontal and the angle of elevation of the sun is 32 degrees, find the height of the tree.
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
based on my diagram, i believe your solution is that the height of the tree is 36.86067667 feet.

here's my diagram.
explanations are below the diagram.

angle DCA is the angle of inclination of the hill to the horizontal.
angle ADC is the angle of elevation of the sun.
the sun if the unmarked dot on the top right of the diagram.
i should have given it a letter but i forgot.
it didn't figure in the measurements anyway, once the angle of elevation was determined to be where it was.

the height of the tree is side F.
the length of the shadow on the hill is side G.
the length of the shadow from the top of the tree to the point on the hill where the shadow ends is side E.

point B is the top of the tree.
point C is the bottom of the tree.

angle DAC is equal to 136 degrees because the sum of the angles of triangle DAC is 180 degrees and the other two angles are 32 and 12 degrees.
32 + 12 = 44 + 136 = 180.

angle BAC is equal to 44 degrees because it is supplementary to angle DAC, which is 136 degrees.
136 + 44 = 180.

angle BCD = 78 degrees because it is complementary to angle ACD, which is 12 degrees.
12 + 78 = 90.

angle ABC = 58 degrees because the sum of the angles of triangle ABC is equal to 180 degrees and the other two angles are 44 degrees and 78 degrees.
44 + 78 + 58 = 180.

the triangle of interest is triangle ABC.

the length of side G is given as 45 feet.

we can use the law of sines to find the length of the other sides of the triangle.

the law of sines states:

G / sin(58) = E / sin(78) = F / sin(44)

since we know that G is equal to 45 feet. then we get:

G / sin(58) = 45 / sin(58) = 53.06302815.

this makes E / sin(78) = 53.06302815.
solve for E to get:
E = 53.06302815 * sin(78) = 51.90347367 feet.

this also makes F / sin(44) = 53.06302815.
solve for F to get:
F = 53.06302815 * sin(44) = 36.86067667 feet.

F is the height of the tree.
that is your solution.
the height of the tree is 36.86067667 feet.