Consider the following for the two (02) items that follow: The top...

Let $MN = h$ be the height of the tower.

The distances of points $P$, $Q$, and $R$ from the foot of the tower $N$ are given by:

$PN = h \cot 30^{{}^{\circ}} = h\sqrt{3}$

$QN = h \cot 45^{{}^{\circ}} = h$

$RN = h \cot 60^{{}^{\circ}} = \dfrac{h}{\sqrt{3}}$

Given that $P$, $Q$, $R$ lie on a straight line passing through $N$, the distance $QR$ is:

$QR = QN - RN$

$b = h - \dfrac{h}{\sqrt{3}}$

$b = h \left(\dfrac{\sqrt{3}-1}{\sqrt{3}}\right)<br><br>$h = $\dfrac{b\sqrt{3}}{\sqrt{3}-1}<br><br>Rationalizing$ the denominator:

h = $\dfrac{b\sqrt{3}(\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)}<br><br>$h = $\dfrac{b(3+\sqrt{3})}{3-1}<br><br>$h = \left(\dfrac{3+\sqrt{3}}{2}\right)$b$

Answer: \left(\dfrac{3+\sqrt{3}}{2}\right)b