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Question from Mike, a parent:

I was reviewing this question and answer:
mathcentral.uregina.ca/QQ/database/QQ.02.06/phil1.html

But I have trouble with this part:
Now if we express the radius of the inside circle as r and the outside circle's radius is R, then this means r/R is 911/1728. But earlier we said that the outside radius R is simply w more than the inside radius r, so R = r + 282. That means that r/R = r/(r + 282). Now we can simply solve the equation for r:
r/(r+282) = 911/1728
This means r = 314 mm (with rounding).

Can I get more detail on the method to solve for r?

Thank you,
Mike

Hi Mike,

From the diagrams in Stephens solution you can see that the radius R of the outside circle is the radius r of the inside circle plus w. That is

R = r + w.

Using the symmetry of a circle he also shows that r/R is 911/1728. Hence

r/(r + w) = 911/1728.

In Phil's problem w = 282 mm so

r/(r + 282) = 911/1728.

At this point multiply each side by (r + 282) to obtain

(r + 282)× r/(r + 282) = (911/1728) × (r + 282)

r = (911/1728) × (r + 282).

Now multiply each side by 1728 and the equation becomes

1728 × r = 911 × (r + 282).

Thus

1728 × r = 911 × r + 911 × 282

1728 × r - 911 × r = 911 × 282.

817 × r = 911 × 282

and finally

r = (911 × 282)/817 = 314 mm.

I hope this helps,
Penny

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