Activity 4.2.4.
Surveyors are trying to determine the height of a hill relative to sea level. First, they choose a point to take an initial measurement with a sextant that shows the angle of elevation from the ground to the peak of the hill is \(19^\circ\text{.}\) Next, they move \(1000\) feet closer to the hill, staying at the same elevation relative to sea level, and find that the angle of elevation has increased to \(25^\circ\text{,}\) as pictured in the given diagram. We let \(h\) represent the height of the hill relative to the two measurements, and \(x\) represent the distance from the second measurement location to the “center” of the hill that lies directly under the peak.
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(a)
Using the right triangle with the \(25^\circ\) angle, find an equation that relates \(x\) and \(h\text{.}\)
(b)
Using the right triangle with the \(19^\circ\) angle, find a second equation that relates \(x\) and \(h\text{.}\)
(c)
Our work in (a) and (b) results in a system of two equations in the two unknowns \(x\) and \(h\text{.}\) Solve each of the two equations for \(h\) and then substitute appropriately in order to find a single equation in the variable \(x\text{.}\)
(d)
Solve the equation from (c) to find the exact value of \(x\) and determine an approximate value accurate to \(3\) decimal places.
(e)
Use your preceding work to solve for \(h\) exactly, plus determine an estimate accurate to \(3\) decimal places.
(f)
If the surveyors’ initial measurements were taken from an elevation of \(78\) feet above sea level, how high above sea level is the peak of the hill?
