Find the height of the mountain to the nearest foot.

Question

Bob is driving along a straight and level road straight toward a mountain.  At some point on his trip, he
measures the angle of elevation to the top of the mountain and finds it to be 24° 10′. He then drives 1 mile
(1 mile = 5280 ft) more and measures the angle of elevation to be 33° 52′.  Find the height of the mountain to
the nearest foot. What formula would you use?

Answer 1

Hi Ellie:

Here's how I would do it: first, draw a diagram (very important)

Label the first point on the road, the second point B, the base of the mountain C and the top of the mountain D.

Then ∠DAB = 24°10' and ∠DBC = 35º52'

The height of the mountain is given by side CD in ΔBCD.

To find CD, we need one more side in ΔBCD, namely, BD.

Now, BD is also in ΔABD. Aha!

If we can find ∠ADB, we can use the sine rule to find side BD.

Now, ∠DBC is the exterior angle of ΔABD => ∠DBC = ∠DAB + ∠ADB (exterior ∠ = sum of interior opp ∠s)

=> 33°52' = 24°10' + ∠ADB

=> ∠ADB = 33°52' - 24°10'

= 9°42'

∴ in ΔABD, AB / sin ∠ADB = BD / sin ∠DAB

=> 5280 / sin 9º42' = BD / sin 24º10'

=> BD = 5280 sin 24º10' / sin 9º42

Finally, ht of mtn, CD = BD sin 33º52'

= 5280 sin 24º10' sin 33º52' / sin 9º42

You can do the calculator math!