SOLUTION: A water tower 30 m tall is located at the top of a hill. From a distance of 120 m down the hill it is observed that the angle formed between the top and base of the tower is 8°.

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Question 1171451: A water tower 30 m tall is located at the top of a hill. From a distance of 120 m down the hill it is observed that
the angle formed between the top and base of the tower is 8°. Find the angle of inclination of the hill; that is, the
angle that the slope of the hill makes with the horizontal.
Answer by ikleyn(52772) About Me  (Show Source):
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A water tower 30 m tall is located at the top of a hill. From a distance of 120 m down the hill
it is observed that the angle formed between the top and base of the tower is 8°.
Find the angle of inclination of the hill;
that is, the angle that the slope of the hill makes with the horizontal.
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Let B be the observation point; let point C be the base of the tower and let point D be the top of the tower.

Let  'a'  be the angle (measured in degrees) of inclination of the hill.

Then the direction from the observation point to the top of the water tower is (a+8°).


Consider right angled triangles ABC and ABD, where AB is the horizontal line from point B.


Write the sine law equation for triangle BCD

    30%2Fsin%288%5Eo%29 = 120%2Fsin%28CDB%29.


From this equation

    sin(CDB) = %28120%2Asin%288%5Eo%29%29%2F30 = 4%2Asin%288%5Eo%29 = 4*0.1391731 = 0.5566924.


Hence, angle CDB is  arcsin(0.5566924) = 33.83  degrees  (rounded).

It implies that the angle ABD is 90° - 33.83° = 56.17 degrees.


Angle 'a' is 8 degrees less than angle ABD, i.e. 56.17 - 8 = 48.17 degrees.


ANSWER.  The hill makes the angle of 47.17 degrees with the horizontal.

Solved.