SOLUTION: Standing on one back of a river flowing north, Mary notices a tree on the opposite bank at a bearing of 115.45o. John is on the same bank as Mary, but 428.3 meters away. John not

Algebra ->  Trigonometry-basics -> SOLUTION: Standing on one back of a river flowing north, Mary notices a tree on the opposite bank at a bearing of 115.45o. John is on the same bank as Mary, but 428.3 meters away. John not      Log On


   

Question 1162993: Standing on one back of a river flowing north, Mary notices a tree on the
opposite bank at a bearing of 115.45o. John is on the same bank as Mary, but
428.3 meters away. John notices that the bearing of the tree is 45.57o. The
two banks are parallel. What is the distance across the river?
Found 3 solutions by josgarithmetic, Edwin McCravy, ikleyn:
Answer by josgarithmetic(39630) About Me  (Show Source):
You can put this solution on YOUR website!
Draw and label the triangle:
J, point where is John
M, point where is Mary
T, the tree on opposite bank
B, point where directly accross river where tree is located
JM=428.3
x=JB
TB, the distance accross river

That forms a triangle JMT.
Interior angle at M is 64.55 degrees.


Question would then be asking for length TB.

system%28TB%2Fx=tan%2845.57%29%2CTB%2F%28428.3-x%29=tan%2864.55%29%29
Solve the system to find TB.

A different method may be possible.

Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!
So that I will not be doing your work for you, I will change the numbers
and work another problem exactly like yours.  So instead of your problem I
will do this one instead:
Standing on one back of a river flowing north, Mary notices a tree on the
opposite bank at a bearing of 123.86°. John is on the same bank as Mary, but
516.4 meters away. John notices that the bearing of the tree is 48.23°. The
two banks are parallel. What is the distance across the river?
 



Bearing angles are taken swinging clockwise from due north, 
which is "straight upward".

Mary is at M. John is at J.  The tree is at T. We want to
know OT denoted by x, the distance across the river. 

MJ = 516.4 meters, which is the sum MO+OJ, denoted by y+z,

So y + z = 516.4

We have two right triangles, MOT and JOT 

Angle OMT = 180°-123.86° = 56.14°

Since tangent = opposite/adjacent

tan%2856.14%5Eo%29=x%2Fyorx=y%2Atan%2856.14%5Eo%29

tan%2848.23%5Eo%29=x%2Fzorx=z%2Atan%2848.23%5Eo%29

Set the expressions for x equal to each other:

y%2Atan%2856.14%5Eo%29=z%2Atan%2848.23%5Eo%29

So we have the system of two equations in two unknowns:

system%28y%2Bz=516.4%2Cy%2Atan%2856.14%5Eo%29=z%2Atan%2848.23%5Eo%29%29

Solve the first equation for z:

y=516.4-z

Substituting in the other equation:

%28516.4-z%29%2Atan%2856.14%5Eo%29=z%2Atan%2848.23%5Eo%29

%28516.4-z%29%2A1.490403551=z%2A1.119618398

769.6443937-1.490403551z=1.119618398z

769.6443937=2.610021949z

769.6443937%2F2.610021949=z

294.9904296=z

Then substitute that for z in

x=z%2Atan%2848.23%5Eo%29

x=294.9904296%2Atan%2848.23%5Eo%29

x=294.9904296%2Atan%2848.23%5Eo%29

x=330.1535541

Now use your numbers and do the exact same steps.

Edwin

Answer by ikleyn(52921) About Me  (Show Source):
You can put this solution on YOUR website!
,

Standing on one highlight%28cross%28back%29%29 bank of a river . . .