91Ó°ÊÓ

Given that while taking a ride in a hot-air balloon in Napa Valley, Francisco wonders how high he is. To find out, he chooses a landmark that is to the east of the balloon and measures the angle of depression to be 54°. A few minutes later, after traveling 100 feet east, the angle of depression to the same landmark is determined to be 61°.

We can draw a figure of the given problem as:-

We have to find the height of the balloon $h.$

So,

In $△ABC,∠B=90°,∠C=54°$

$$\begin{gathered} \tan (54°)=\frac{AB}{BC} \\ AB=\tan (54°)×BC \\ x=\frac{h}{\tan (54°)}-----\left(1\right) \end{gathered}$$

Now, in $△ABD,∠B=90°,∠D=61°$

$$\begin{gathered} \tan (61°)=\frac{AB}{BD} \\ \tan (61°)×BD=AB \\ BD=\frac{h}{\tan (61°)} \\ x-100=\frac{h}{\tan (61°)}-----\left(2\right) \end{gathered}$$

Now, putting the value of $x$from (1 ) into (2),

$$\begin{gathered} \frac{h}{\tan (54°)}-100=\frac{h}{\tan (61°)} \\ \frac{h}{1.376}-100=\frac{h}{1.804} \\ 0.726h-100=0.554h \\ 0.726h-0.554h=100 \\ 0.172h=100 \\ h=\frac{100}{0.172} \\ h=581.40feet \end{gathered}$$

So, the height of the balloon is 581.40 feet.