# Calculus - Related Rates

The length of a rectangle is increasing at a rate of 3 inches per minute and the width is decreasing at a rate of 2 inches per minute. At the moment when the length is 8 inches and the width is 6 inches, how fast is the angle formed by the length and the diagonal changing? Give your answer in radians per minute.

Basically, note that if the length of the rectangle is $l$ and the width $w$, we have: $$\frac{dl} {dt} =3\tag 1$$ $$\frac{dw} {dt} =-2 \tag 2$$ Now, note that if $\theta$ is the angle between the length and the main diagonal, a simple diagram yields the observation: $$\theta = \arctan\left(\frac{w} {l}\right) \tag 3$$
I leave it to you to differentiate $(3)$ and substitute the appropriate values of $l$ and $w$.
• $x$=length $\implies \frac{\partial x}{\partial t}=3$
• $y$=width $\implies \frac{\partial y}{\partial t}=-2$
• $z$=angle $\implies z=\arctan\frac{y}{x}$