hot-air balloon is floating above a straight road. To calculate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be 26∘ and 30∘.
How high (in feet) is the ballon?

In the event when the line of sight is directed downhill from the horizontal line, an angle is created with it. Angle of depression is the angle produced between the horizontal line and the observer's line of sight when the object being observed is below the level of the observer.

Let's give the balloon's height above the earth and the distance it is from the first milepost horizontally a name.

We can state the following if the angle of this milepost's depression is 24 degrees:

Tan(24)=[tex]\frac{h}{b}[/tex]

b = [tex]\frac{h}{tan(24)}[/tex]

We can state the following for the following milepost as there will be (b+1)miles horizontal distance of and a 20 degree depression at that location.

tan(20)=[tex]\frac{h}{b+1}[/tex]

b +1 = [tex]\frac{h}{tan(20)}[/tex]

Therefore, substituting equation 1

[tex]\frac{h}{tan(24)}[/tex] = [tex]\frac{h}{tan(20)} -1[/tex]

1 = [tex]\frac{h}{tan(20)}[/tex] - [tex]\frac{h}{tan(24)}[/tex]

1 = h× ( [tex]\frac{h}{tan(20)}[/tex] - [tex]\frac{h}{tan(24)}[/tex] )

h = 1.99 miles

Consequently, the height of balloon is 1.99 miles above the ground.

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