# Thread: Rate of Change Problem

1. ## Rate of Change Problem

Hi, I was just wondering if someone could tell me if I am doing the following problem correctly?

The vertices of a square are moving away from the centre at a rate of 2 cm/s. How fast is the area of the inscribed circle changing when the side of the square is 20 cm?

I let r=10 cm and plugged it into the derivative of the area of a circle... is this what I am supposed to do?

Thanks for any help

2. r - radius of inscribed circle at start
a - side of square at start;

$\displaystyle r=a/2$, you spotted that well.

Now "The vertices of a square are moving away from the centre at a rate of 2 cm/s ", what that means actually?

Spoiler:
That means the diagonal of the rectangle is increasing by 2x of both sides, where x is s, as you stated

Here is the calculation of the side of the rectangle:

Spoiler:

$\displaystyle 2x+a*\sqrt{2}+2x=4x+a*\sqrt{2}$

z - the side of rectangle after s passed

$\displaystyle (4x+\sqrt{2}*a)^2=2z^2$
$\displaystyle (4x+\sqrt{2}*a)=\sqrt{2}z$
$\displaystyle 4x/\sqrt{2} + a = z$

Now what is r after?

Spoiler:

$\displaystyle r_{after}=2x/\sqrt{2} + a/2$

Finally,

$\displaystyle A=r^2_{after}*\pi$

Regards.