Right Angle Trigonometry

Question

In traveling across flat land, you see a mountain directly in front of you. Its angle of elevation (to the peak) is 3.5°. After you drive 11 miles closer to the mountain, the angle of elevation is 9° (see figure). Approximate the height of the mountain. (Round your answer to one decimal place.)

Answer 1

Let h = height (in miles) of mountain and let x = distance (in miles) from the base of the mountain to your original position.

 

Then, tan3.5° = h/x.  So, h = xtan3.5°

 

         tan9° = h / (x-11).  So, h = (x-11)tan9°

 

Therefore, (x-11)tan9° = xtan3.5°

 

                xtan9° - 11tan9° = xtan3.5°

 

                xtan9° - xtan3.5° = 11tan9°

 

                x(tan9° - tan3.5°) = 11tan9°     x = 17.92 miles

 

So, h = xtan3.5° = 1.096 miles = 5786.9 ft

 

Answer 2

Draw a figure and label the points as follows:

A is the starting pont, B is the top of the mountain, C is the point where the perpendicular from B intersects the land and D is the second point of observation.

∠ADB = 171° because it is supplemental to 9°.

That makes ∠ABD = 5.5°.

The hypotenuse of ΔBCD can be found from using the law of sines in ΔABD.

The height of the mountain = the hypotenuse just found * the sin 9°.

 

I will leave the numbers to you.