TITLE: Related Rates - Growth of the side of a triangle
QUESTION [1 upvotes]: recently I saw a question on a book but I personally don't agree with the answer. I would appreciate if you could help me achieve the final result. It follows:
Two sides of a triangle measure $12m$ and $15m$. 
The angle between the two sides grow at the rate of $2^o/min$. How fast is the third side growing when the angles between the sides that have constant size is $60^o$ ?
Thank you,
Best Regards,
Bruno

REPLY [1 votes]: Let $x=x(t)$  be the third side, and $\theta$ the angle between the first two sides. Using the Law of Cosines,
when $\theta = 60^\circ$
$$x^2 = 15^2+12^2- 2 \cdot 15 \cdot 12 \cdot \cos(60^\circ)$$
$$x = 3 \cdot \sqrt{21}$$
For the Related Rate,
$$[x^2]' = 0-2 \cdot 15 \cdot 12 [ \cos \theta]'$$
$$2x \cdot x' = 2 \cdot 15 \cdot 12 \cdot \sin(\theta) \cdot [\theta]'$$
Since $[\theta]' = 2 \cdot \pi /180$
$$2 (3 \cdot \sqrt{21}) \cdot x' = 2 \cdot 15 \cdot 12 \cdot (\sqrt{3}/2) \cdot (2 \cdot \pi /180) $$
$\therefore x' = \pi \cdot \frac{\sqrt{7}}{21}$
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