A flagpole is located at the edge of a sheer y = 70-ft cliff at the bank of a river of width x = 40 ft. See the figure below. An observer on the opposite side of the river measures an angle of 9° between her line of sight to the top of the flagpole and her line of sight to the top of the cliff. Find the height of the flagpole.
I would solve this using tangents. Let h be height of flagpole. Set up 2 right triangles, each with a base of 40. The larger triangle has height of "h+70" Smaller triangle has height of 70.
Now write the tangent ratios: [tex]tan A = \frac{h+70}{40} , tan B = \frac{70}{40} [/tex]
Note: A-B = 9 To solve for h we need to use the "Difference Angle" formula for Tangent [tex]tan (A-B) = \frac{tanA - tanB}{1+tan A tan B}[/tex] Plug in what we know: [tex]tan(9) = \frac{ \frac{h+70}{40} - \frac{70}{40}}{1+ (\frac{h+70}{40})(\frac{7}{4})} [/tex] [tex]tan (9) = \frac{ \frac{h}{40}}{ \frac{7h +650}{160}} = \frac{4h}{7h+650} [/tex] [tex]h = \frac{650 tan(9)}{4-7 tan(9)}[/tex] [tex]h = 35.6[/tex]
