Question\#5 Shari and Leo are standing 36 m apart and facing each other. If Shari looks upward, the angle of elevation of a hot air balloon in the sky is $\frac{\pi}{4}$ radians and from Leo the angle of elevation is $\frac{\pi}{3}$ radians. Calculate the height of the hot air balloon from the ground.

Studdy Solution
Solve for $x$:
$x = \sqrt{3}(36 - x)$
$x = 36\sqrt{3} - \sqrt{3}x$
$x + \sqrt{3}x = 36\sqrt{3}$
$x(1 + \sqrt{3}) = 36\sqrt{3}$
$x = \frac{36\sqrt{3}}{1 + \sqrt{3}}$
Now, substitute back to find $h$:
$h = x = \frac{36\sqrt{3}}{1 + \sqrt{3}}$
Rationalize the denominator:
$h = \frac{36\sqrt{3}(1 - \sqrt{3})}{(1 + \sqrt{3})(1 - \sqrt{3})}$
$h = \frac{36\sqrt{3} - 108}{1 - 3}$
$h = \frac{36\sqrt{3} - 108}{-2}$
$h = 54 - 18\sqrt{3}$
The height of the hot air balloon from the ground is:
$$\boxed{54 - 18\sqrt{3}} \, \text{meters}$$