Question 245007
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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 *[tex \Large x] 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 *[tex \Large x\ -\ 105].  Let *[tex \Large y] 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:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \tan(15.8)\ =\ \frac{y}{x}]


and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \tan(21.4)\ =\ \frac{y}{x-105}]


Using a tangent table or a scientific calculator we can determine that:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \tan(15.8)\ \approx\ 0.283]


and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \tan(21.4)\ \approx\ 0.392]


So:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ \approx\ 0.283x]


and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ \approx\ 0.392(x\ -\ 105)]


Solve the system by setting the two expressions that are equal to *[tex \Large y] equal to each other, i.e. solve by substitution.  One decimal place precision in your answer should be more than sufficient.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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