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A hot air balloon is floating above a straight road. To estimate 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 20 degrees and 24 degrees. How high is the balloon?

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dfrancolopera

Answer:

[tex]h=1.99 miles[/tex]

Step-by-step explanation:

Let's call [tex]h[/tex] the balloon's height above the ground and [tex]b[/tex] the horizontal distance between the balloon and the first milepost.

If the angle of the depression of this milepost is 24 degrees, we can say the following:

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

[tex]b=\frac{h}{tan(24)}[/tex](equation 1)

For the next milepost, the horizontal distance between the balloon and tihs milepost will be [tex](b+1)miles[/tex] and the angle of depression will be 20 degrees, so we can say the following

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

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

[tex]b=\frac{h}{tan20}-1[/tex]

then, replacing from equation 1

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

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

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

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

Resolving, we'll find the value of [tex]h[/tex]

[tex]h=1.99 miles[/tex]

So, the balloon's height above the ground is 1.99miles