Two ﬁre towers are 30 kilometers apart, tower
A being due west of tower B. A house ﬁre
is spotted from the towers, and the bearings
from A and B are N 72◦ E and N 52◦ W, respectively.

Find the distance d of the ﬁre from the line
segment AB.
Answer in units of km

Question

Two ﬁre towers are 30 kilometers apart, tower
A being due west of tower B. A house ﬁre
is spotted from the towers, and the bearings
from A and B are N 72◦ E and N 52◦ W, respectively.

Find the distance d of the ﬁre from the line
segment AB.
Answer in units of km

anonymous · Answer

attachment

anonymous · Answer

tan 38  = d / x
tan 18 = d / (30-x)

solve these two equations simultaneously to find values of x and d.

anonymous · Answer

from first equation:
x = d / tan 38

plug this into second equation
and solve for d

anonymous · Answer

tan18= 30/tan38 ?

anonymous · Answer

tan18 =  d / ( 30  -  d/tan38)
 tan18  =  d  /   (30tan38 - d) / tan38

tan 18 =  d  *   tan 38 
                      -------
                     30tan38 - d

d tan38  =  30tan38tan18 - dtan18

d =   30tan38tan18
        ------------
        tan38 + tan 18

   =   7.61565  /  1.10621
 =  6.88 km

anonymous · Answer

the algebra's a bit messy