Jeff K. answered 06/08/20
Together, we build an iron base of understanding
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'
∴ 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!