0

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

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.

### 2 Answers by Expert Tutors

Nataliya D. | Patient and effective tutor for your most difficult subject.Patient and effective tutor for your mos...
0

B
•
|   \
| γ   \
|         \
|            \
|               \
|                  \
A  •  \                  \
|  \ α   \             \
|     \       \           \
|        \        \          \
|           \         \         \
|              \           \       \
|                 \            \      \
C  •                  •   CANYON \   •  D
E

< C = 90o ;
< BDC = 180o - (90o + 48o) = 42o
< CAD = 53o - 13o = 40o
< AEC = 90o - 40o = 50o
< AED = 180o - 50o = 130o

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

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

then  <ADB = 180o - (48o + 127o) = 5o

---------  =  ----------
sin 48o        sin 5o

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

---------  =  ---------
sin 130o       sin 13o

ED ≈ (3410.7 * sin 13o) / sin 130o = 1001.56 feet ≈ 1002 ft
Andre W. | Friendly tutor for ALL math and physics coursesFriendly tutor for ALL math and physics ...
5.0 5.0 (3 lesson ratings) (3)
0
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.