Solve using systems of equations need to find the width of the canyon to the nearest foot

Question

A forest service helicopter needs to determine the width of a deep canyon. While hovering they measure the angle gamma =48 degrees at position B then descend 400ft to position A and make two measurements alpha =13 (the measure of angle EAD) beta =53 degrees (the measure of angle CAD) determine the width of the canyon to the nearest foot.

Answer 1

     B
     •
     |   \         
     | γ   \  
     |         \
     |            \
     |               \
     |                  \
 A  •  \                  \
     |  \ α   \             \
     |     \       \           \
     |        \        \          \
     |           \         \         \
     |              \           \       \
     |                 \            \      \
 C  •                  •   CANYON \   •  D  
                        E          
 
< C = 90^o ;  
< BDC = 180^o - (90^o + 48^o) = 42^o
< CAD = 53^o - 13^o = 40^o
< AEC = 90^o - 40^o = 50^o
< AED = 180^o - 50^o = 130^o

I think the rational way will be to use the law of sines.

Consider the Δ BAD:

<CBD = 48^o (given)
<BDA = 180^o - <CAD =
             189^o - 53^o = 127^o  
(<CAD = 53^o is given)

then  <ADB = 180^o - (48^o + 127^o) = 5^o

  AD             400 ft
---------  =  ---------- 
 sin 48^o        sin 5^o  


AD = (400 * sin 48^o) / sin 5^o = 3410.65225 ft

Consider the Δ EAD :    

   AD              ED
---------  =  ---------
sin 130^o       sin 13^o

ED ≈ (3410.7 * sin 13^o) / sin 130^o = 1001.56 feet ≈ 1002 ft

Answer 2

Use tangent = opposite /adjacent for the three right triangles with angles γ=48°, β=53°, and β-α=40°:

 

tan 48 = (CE+ED)/(400+AC)

tan 53 = (CE+ED)/AC

tan 40 = CE/AC

 

Eliminate CE+ED from the first two equations and solve for AC:

 

(400+AC) tan 48 = AC tan 53

AC = (400 tan 48)/(tan 53 - tan 48) = 2053

 

Find CE from the last equation:

 

CE = AC tan 40 = 2053 tan 40 = 1722

 

Now find ED, which is the width of the canyon, from the 2nd equation:

 

ED = AC tan 53 - CE = 2053 tan 53 - 1722 = 1000 feet.