SOLUTION: A surveyor measures the angle of elevation of a hill at 15.8 degrees. She then moves 105m closer, on level ground, and measures the angle of elevation as 21.4 degrees. Calculate th
Question 245007: A surveyor measures the angle of elevation of a hill at 15.8 degrees. She then moves 105m closer, on level ground, and measures the angle of elevation as 21.4 degrees. Calculate the height of hill using system of equations. Answer by solver91311(24713) (Show Source):
Of course we have to assume that the measurements were taken from the surface of a flat plane that extends under the entirety of the hill. We also need to assume that the height of the hill is the measurement of the line segment that is normal to the plane and ends at the highest point of the hill.
Let represent the measure of the line segment from the intersection of the height segment and the plane and the point on the plane from which the first angle measurement was taken. Then the measure of the segment from the intersection of the height segment with the plane and the point from which the second measurement was taken must be represented by . Let represent the measure of the height segment.
For each of the measurements, the endpoints of the height segment and the endpoints of the distance to the measurement point segment form a right triangle. The definition of the tangent function tells us that the tangent of an angle is equal to the ratio of the opposite side to the adjacent side, hence we can write:
and
Using a tangent table or a scientific calculator we can determine that:
and
So:
and
Solve the system by setting the two expressions that are equal to equal to each other, i.e. solve by substitution. One decimal place precision in your answer should be more than sufficient.
