Can someone help me answer this problem

[nch]

I saw this problem and I thought it would be cool if I could figure it out. Unfortunately I can't figure out how to do it . Can one of you guys solve the problem and put the explanation as a reply on this thread? Thanks.


Here is the problem:

You have 2 circles, one with the radius r and the other one with Radius R. You wish for the difference in the areas of these two circles to be less than or equal to 5 pi. If r+R=10, what is the maximum difference in the lengths of the radii?

 

[soroban]

Hello, nch!

You have 2 circles, one with the radius r and the other one with radius R.
You wish for the difference in the areas of these two circles to be less than or equal to 5 pi.
If r + R = 10, what is the maximum difference in the lengths of the radii?

Let $\displaystyle R$ be the radius of the larger circle.
. . Its area is: $\displaystyle \pi R^2$

Let $\displaystyle r$ be the radius of the smaller circle.
. . Its area is: $\displaystyle \pi r^2$

We have: . . .$\displaystyle \pi R^2 - \pi r^2 \;\le\;5\pi$

Divide by $\displaystyle \pi\!:\quad\: R^2 - r^2 \;\le\;5$

Factor: .$\displaystyle \underbrace{(R+r)}_{\text{This is 10}}(R-r) \;\le\;5$

. . . . . . . . . .$\displaystyle 10(R-r) \;\le\;5$

. . . . . . . . . . . . . $\displaystyle R-r \;\le\;\frac{1}{2}$