Find the altitude if the plane

Question

Altitude The angles of elevation to an air plane from two points A and B on level ground are 71° and 88°,respectively .The points A and B are 5.5 miles apart and the airplaine is east if both points in the same vertical plane. Find the altitude of the plane. 

Answer 1

If the plane and the two points are in the same vertical plane, the solution will be:

Calculate the angle formed by the visual lines at the plane point; It will be 180 - 88 - 71 = 21 degrees

According to the sinus law we may write:

5.5(dist between points)/sin 21 = a(opposite to angle A)/ sin 71 = b/sin 88

solving for a and b we obtain:

a = 14.51 miles

b = 15.33 miles

Then we can form tho rectangles triangles, for one the vertical side, forms 90 degrees with the soil then will be the altitude.

Sinus theorem; 14.1/1 = h/sin 88 solving for h = 14.49 miles

To check this answer we can do it in the other triangle:

15.33/1(sin 90) = h/ sin 71, solving for h gives 14.49. so CHECK!!!!!!!!!!!!

 

Answer 2

let point C be where the plane is

you have triangle ABC

angle A is 71º

angle B is 88º (outside triangle ABC)

angle B inside the triangle is 92º (180-88)

angle C is 17º (71+92+17=180)

using the law of sines:

sin(17)/5.5=sin(71)/BC

0.29237/5.5=0.94552/BC

0.29237(BC)=5.5(0.94552)

0.29237(BC)=5.20036

BC=5.20036/0.29237

BC=17.7869 miles

now you have another triangle CBD where CD is the altitude of the plane

using the law of sines again:

sin(90)/BC=sin(88)/CD where CD is the altitude

1/17.7869=sin(88)/CD

CD=17.7869(0.99939)

CD=17.77605 miles

Answer 3

•  Plane

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| altitude(a)

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•________X______________•B______5.5m________•A

 

 

TanA=    a    

            x+5.5

 

TanB=   a  

           x

 

Tan(71°)(x+5.5) = a

Tan(88°)(x) = a

 

Tan(71°)(x+5.5)=Tan(88°)x

 

(x)Tan(71°)+(5.5)Tan(71°) = (x)Tan(88°)

 

(5.5)Tan(71°) = (x)Tan(88°)-(x)Tan(71°)

 

(5.5)Tan(71°) = x(Tan88°-Tan71°)

 

x =  (5.5)Tan71°   

     Tan88°- Tan71°

 

After finding x, plug into:   a = (x)Tan88°   or    a = Tan(71°)(x+5.5)     (the former is easier)