Question

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.

Answer 1

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:
$tan A = \frac{h+70}{40} , tan B = \frac{70}{40}$

Note: A-B = 9
To solve for h we need to use the "Difference Angle" formula for Tangent
$tan (A-B) = \frac{tanA - tanB}{1+tan A tan B}$
Plug in what we know:
$tan(9) = \frac{ \frac{h+70}{40} - \frac{70}{40}}{1+ (\frac{h+70}{40})(\frac{7}{4})}$
$tan (9) = \frac{ \frac{h}{40}}{ \frac{7h +650}{160}} = \frac{4h}{7h+650}$
$h = \frac{650 tan(9)}{4-7 tan(9)}$
$h = 35.6$