increments and differentiation to determine related rates

Question

In an isosceles triangle with sides 20 inches long and a vertex angle of $60^\circ$, the angle is closing in at $2^\circ/min$. At what rate should the sides be changing to keep the area of the triangle constant (at this instant)? indicate the type of change.

Answer 1

Each quantity there is a function of time including the area. $x$ is one of the two legs of the triangle. The area of the triangle: $A=hb$. Express $h$ and $b$ in terms of $x$ and $\theta$: $$\cos{\theta}=\frac{h}{x}\implies h=x\cos{\theta}\\ \sin{\theta}=\frac{b}{x}\implies b=x\sin{\theta}$$ Therefore, we can express the area the following way: $$A=x^2\cos{\theta}\sin{\theta}=\frac{1}{2}x^2 2\cos{\theta}\sin{\theta}=\frac{1}{2}x^2\sin{2\theta}$$

Saying an area is not changing with time implies that it's a constant function of time. The derivative of a constant function is zero. Implicitly differentiate both sides with respect to time and isolate $\frac{dx}{dt}$—the rate of change you're looking for:

$$ \frac{dA}{dt}=\frac{d}{dt}\left[\frac{1}{2}x^2\sin{2\theta}\right]\implies\\ 0=x\frac{dx}{dt}\sin{2\theta}+x^2\cos{2\theta}\frac{d\theta}{dt}\implies\\ x\frac{dx}{dt}\sin{2\theta}=-x^2\cos{2\theta}\frac{d\theta}{dt}\implies\\ \frac{dx}{dt}=-x\cot{2\theta}\frac{d\theta}{dt} $$ Plug in all the known quantities at that instant, keeping in mind that $\theta$ according to the picture is $30^\circ$ which is equivalent to $\pi/6$ in radian measure and the rate of change of the angle $\theta$ should be $-1$ degree per minute or equivalently $-\pi/180$ radians per minute (the negative rate of change indicates that the angle is getting smaller with time) because as the angle decreases by one degree on the right side, the same thing is happening with the other angle on the left side. $$ \frac{dx}{dt}=-20\cdot\cot{\left(2\cdot\frac{\pi}{6}\right)}\cdot\left(-\frac{\pi}{180}\right)=\frac{\pi}{9}\cdot\cot{\left(\frac{\pi}{3}\right)}=\frac{\pi}{9\sqrt{3}}\ in/min $$

The fact that the rate of change is positive indicates that the sides are getting larger (it's also a simple geometric observation).