Trigonometric Right traingle

Question

A woman standing on a hill sees a flagpole that she knows is 45 ft tall. The angle of depression to the bottom of the pole is 14°, and the angle of elevation to the top of the pole is 18°. Find her distance x from the pole. (Round your answer to one decimal place.)

Answer 1

We can basically break this down into 2 triangles having the same horizontal side which is the distance from the pole (x) in feet.

 

The top triangle will have an angle of elevation (baseline is horizontal line of eyesight) of 18^o.  Let the length of the opposite side of this triangle be represented by y.

 

Using tangent:

     tan (18^o) = y/x

Solve for x:  x = y/tan (18^o) = y/0.325 = 3.08y   (equation 1)

 

The bottom triangle will have an angle of depression (baseline is horizontal line of eyesight) of 14^o.  Let the length of the opposite side of this triangle be represented by 45 -y.  Remember total height is 45 ft.
 
Using tangent:
     tan (14^o) = (45-y)/x
Solve for x:  x = (45-y)/tan (14^o) = (45-y)/0.249 = 4.01(45-y)

                   x = 180.45 - 4.01y       (equation 2)

 

Set equation 1 and 2 equal to each other and solve for y:

     180.45 - 4.01y = 3.08y

     180.45 = 7.09y

     y = 25.45 ft

 

From equation 1:

      x = 3.08y = 3.08(25.45)

      x = 78.39 ft

 

Round to one decimal place:   x = 78.4 ft.

 

Answer 2

This problem is a bit tricky.... it seems like you don't have enough information, right? 

 

I wish I could draw you a picture. Try to draw the two triangles that are formed. They are back to back. The two hypotenuses would connect from the right end of my dotted line below to the top and bottom of the vertical dashed lines. 

 

        |

        |.....x.......^18°.....

 ↑     |              14º

        |

 

total height = 45 ft. Top half (in the 18º triangle) can be (45-y) and bottom half can be y. 

 

so we have two relationships: 

tan 18º = (45 - y)/x    and

tan 14º = y/x

 

Use your calculator and get values for Tan 14 = 0.250  and Tan 18 = 0.325

 

so we have 0.325 = (45-y)/x    and     0.250 = y/x 

 

cross multiply and rewrite these in standard form: 

0.325 (x) = 45-y

0.325x +y = 45   and for the other   0.250(x) = y   or    0.250(x) - y = 0 

 

now add these two standard form equations together and the y's cancel and you get 

 

0.575(x) =45

 

x ≈ 78.3 Ft.