caleb and emily are standing 100 yards from each other. caleb looks up at a 45 degree angle to see a hot air balloon. Emily looks yup at a 60 degree angle to see the same hot air ballon. approximate how far is the hot air ballon off the ground?

check the picture below.

thus then

[tex]\bf \begin{cases} y=100-a\\ y=a\sqrt{3} \end{cases}\implies 100-a=a\sqrt{3}\implies 100=a+a\sqrt{3} \\\\\\ 100=a(1+\sqrt{3})\implies \boxed{\cfrac{100}{1+\sqrt{3}}=a} \\\\\\ \textit{what is the altitude \underline{y} then?}\qquad y=100-a \\\\\\ y=100-\cfrac{100}{1+\sqrt{3}}\implies y\approx 63.39745962156~yards[/tex]