Steps to Find Cliff Height

1. Define Variables:

   Let the cliff height be $h$. Let the distances from the cliff base to the observation points be $x$ and $y$. We are given $x + y = 200\text{ m}$.

2. Apply Trigonometric Ratios:

   Using the angles of elevation:

   • For the point with $30^{\circ}$ angle: $\tan(30^{\circ}) = \frac{h}{x}$
   • For the point with $45^{\circ}$ angle: $\tan(45^{\circ}) = \frac{h}{y}$
3. Express Distances ($x$, $y$) in Terms of Height ($h$):
   • $x = \frac{h}{\tan(30^{\circ})} = h \cot(30^{\circ}) = h\sqrt{3}$
   • $y = \frac{h}{\tan(45^{\circ})} = h \cot(45^{\circ}) = h(1) = h$
4. Combine Equations:

   Substitute $x$ and $y$ into the distance equation $x + y = 200$:

   $ h\sqrt{3} + h = 200 $

5. Solve for $h$:

   Factor out $h$:

   $ h(\sqrt{3} + 1) = 200 $

   Isolate $h$:

   $ h = \frac{200}{\sqrt{3} + 1} $

6. Rationalize the Denominator:

   Multiply by the conjugate $(\sqrt{3} - 1)$:

   $ h = \frac{200(\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)} = \frac{200(\sqrt{3} - 1)}{3 - 1} $

   $ h = \frac{200(\sqrt{3} - 1)}{2} $

   $ h = 100(\sqrt{3} - 1)\text{ m} $

The height of the cliff is $100(\sqrt{3} - 1)\text{ m}$.