Answer:
The distance from hot air balloon to the ground [tex]AC=50.1457 ft[/tex].
Step-by-step explanation:
Labelled diagram of given scenario is shown below.
Given that,
An angle of elevation of Hot air balloon by Angie is [tex]40[/tex]°.
When she stepped back [tex]200 ft[/tex] then angle of elevation was [tex]10[/tex]°.
Height of Angie is [tex]5.5 ft[/tex].
To find: How far off the ground is a hot air balloon.
So, from figure
Height of Angie [tex](BC) = 5.5ft[/tex]
In triangle Δ[tex]ABD[/tex],
⇒ [tex]tan {40\si {\degree}} = \frac{AB}{BD}[/tex]
⇒ [tex]AB = tan (40) \times BD[/tex] ....................(1)
Now, In triangle Δ[tex]ABE[/tex]
⇒ [tex]tan (10)= \frac{AB}{200+BD}[/tex]
⇒ [tex]AB = tan(10)\times (200+BD)[/tex]
Here, substituting the value [tex]AB[/tex] from Equation (1) we get,
[tex]tan(10)\times (200+BD)= tan(40)\times BD[/tex]
[tex]tan(10)\times 200+tan(10)\times BD= tan(40)\times BD[/tex]
⇒ [tex]BD\times (tan(40)-tan(10))=tan(10)\times 20[/tex]
⇒ [tex]BD\times 0.6628 = 35.2654[/tex]
⇒ [tex]BD = \frac{35.2654}{0.6628} =53.2067 ft[/tex]
Now, finding the value of [tex]AB[/tex] from equation (1)
[tex]AB= tan(40)\times 53.2067=0.8391\times 53.2067[/tex]
[tex]AB=44.6457 ft[/tex]
Therefore Length of [tex]AC= AB+BC[/tex] = [tex](44.6457 + 5.5) ft[/tex]
[tex]AC=50.1457 ft[/tex].
Hence,
The distance from hot air balloon to the ground [tex]AC=50.1457 ft[/tex].
