Question

a surveyor standing some distance from a mountain, measures the angle of elevation from the ground to the top of the mountain to be 57°2428\. the surveyor then walks forward 1200 feet and measures the angle of elevation to be 65°4729\. what is the height of the mountain? round your solution to the nearest whole foot.

Explanation:

Step1: Convert angles to decimal degrees

First, convert $57^{\circ}24'28''$ and $65^{\circ}47'29''$ to decimal - degrees.
$57^{\circ}24'28''=57 + \frac{24}{60}+\frac{28}{3600}\approx57 + 0.4+0.00778 = 57.40778^{\circ}$
$65^{\circ}47'29''=65+\frac{47}{60}+\frac{29}{3600}\approx65 + 0.78333+0.00806 = 65.79139^{\circ}$

Step2: Set up tangent equations

Let the height of the mountain be $h$ and the initial distance from the surveyor to the base of the mountain be $x$.
We know that $\tan(57.40778^{\circ})=\frac{h}{x}$, so $h = x\tan(57.40778^{\circ})$
Also, $\tan(65.79139^{\circ})=\frac{h}{x - 1200}$, so $h=(x - 1200)\tan(65.79139^{\circ})$

Step3: Equate the two expressions for $h$

$x\tan(57.40778^{\circ})=(x - 1200)\tan(65.79139^{\circ})$
$x\times1.5697=(x - 1200)\times2.2277$
$1.5697x=2.2277x-1200\times2.2277$
$1200\times2.2277 = 2.2277x-1.5697x$
$2673.24=0.658x$
$x=\frac{2673.24}{0.658}\approx4062.67$

Step4: Calculate the height of the mountain

Substitute $x$ into $h = x\tan(57.40778^{\circ})$
$h = 4062.67\times1.5697\approx6377$

Answer:

$6377$