Answer:
63.39 yards.
Step-by-step explanation:
Refer the attached figure
We are given that Caleb and Emily are standing 100 yards from each other i.e. BC = 100
Let BD = x
So, DC = 100-x
We are given that Caleb looks up at a 45° angle to see a hot air balloon i.e. ∠ABD = 45° and Emily looks up at a 60° angle to see the same hot air balloon i.e. ∠ACD = 60°
Let AD be the height of the balloon denoted by h.
In ΔABD
Using trigonometric ratio
[tex]tan \theta = \frac{Perpendicular}{Base}[/tex]
[tex]tan 45^{\circ} = \frac{AD}{BD}[/tex]
[tex]1= \frac{h}{x}[/tex]
[tex]x=h[/tex] ---1
In ΔACD
Using trigonometric ratio
[tex]tan \theta = \frac{Perpendicular}{Base}[/tex]
[tex]tan 60^{\circ} = \frac{AD}{DC}[/tex]
[tex]\sqrt{3}= \frac{h}{100-x}[/tex]
[tex]\sqrt{3}(100-x)=h[/tex] ---2
So, equating 1 and 2
[tex]\sqrt{3}(100-x)=x[/tex]
[tex]100\sqrt{3}-\sqrt{3}x=x[/tex]
[tex]100\sqrt{3}=x+\sqrt{3}x[/tex]
[tex]100\sqrt{3}=x(1+\sqrt{3})[/tex]
[tex]\frac{100\sqrt{3}}{1+\sqrt{3}}=x[/tex]
[tex]63.39=x[/tex]
Thus the height of the balloon is 63.39 yards.
