Related rates question.

Question

Two sides of a triangle have lengths $\sqrt{21}~m$ and $\sqrt{7}~m$. The angle between them is increasing at a rate of $\dfrac{2}{\sqrt{3}}~rad/sec$. How fast is the altitude of the triangle decreasing when the angle between the sides of fixed length is $\dfrac{5\pi}{6}~rad$?

Answer 1

Hint: Let $c$ be the remaining side, then $c = \sqrt{21 + 7 - 2\sqrt{21\cdot 7}\cos \theta} = \sqrt{28 - 14\sqrt{3}\cos \theta}$. We have: $\dfrac{\sqrt{21}\cdot \sqrt{7}\sin \theta}{2} = \dfrac{c\cdot h}{2} \Rightarrow h = \dfrac{7\sqrt{3}\sin \theta}{\sqrt{28-14\sqrt{3}\cos \theta}}$. You can take it from here.