SOLUTION: Express the ratio of the area of the larger circle to the area of the smaller circle in simplest radical form. Radius of larger circle is 2 + sqrt of 3 Radius of smaller circle i

Algebra ->  Surface-area -> SOLUTION: Express the ratio of the area of the larger circle to the area of the smaller circle in simplest radical form. Radius of larger circle is 2 + sqrt of 3 Radius of smaller circle i      Log On


   



Question 824332: Express the ratio of the area of the larger circle to the area of the smaller circle in simplest radical form.
Radius of larger circle is 2 + sqrt of 3
Radius of smaller circle is 2 - sqrt of 3
Advance Thanks to whoever answers :)

Answer by jsmallt9(3759) About Me  (Show Source):
You can put this solution on YOUR website!
The formula for area of a circle is:
A+=+pi%2Ar%5E2
So the area for the larger circle will be:
A%5BL%5D+=+pi%2A%282%2Bsqrt%283%29%29%5E2
Using the %28a%2Bb%29%5E2+=+a%5E2%2B2ab%2Bb%5E2 pattern to square the radius:
A%5BL%5D+=+pi%2A%28%282%29%5E2%2B2%282%29%28sqrt%283%29%29%2B%28sqrt%283%29%29%5E2%29
Simplifying...
A%5BL%5D+=+pi%2A%284%2B4sqrt%283%29%2B3%29
A%5BL%5D+=+pi%2A%287%2B4sqrt%283%29%29

Repeating this for the smaller circle:
A%5BS%5D+=+pi%2A%282-sqrt%283%29%29%5E2
Using the %28a-b%29%5E2+=+a%5E2-2ab%2Bb%5E2 pattern to square the radius:
A%5BS%5D+=+pi%2A%28%282%29%5E2-2%282%29%28sqrt%283%29%29%2B%28sqrt%283%29%29%5E2%29
Simplifying...
A%5BS%5D+=+pi%2A%284-4sqrt%283%29%2B3%29
A%5BS%5D+=+pi%2A%287-4sqrt%283%29%29

Now we find the ratio of these areas:
A%5BL%5D%2FA%5BS%5D+=+%28pi%2A%287%2B4sqrt%283%29%29%29%2F%28pi%2A%287-4sqrt%283%29%29%29
Simplifying... The pi's cancel:
A%5BL%5D%2FA%5BS%5D+=+%287%2B4sqrt%283%29%29%2F%287-4sqrt%283%29%29
This may be an acceptable answer.

But it does have a square root in the denominator. Usually denominators should be rationalized:

Using the %28a%2Bb%29%5E2 pattern on top and the %28a%2Bb%29%28a-b%29=a%5E2-b%5E2 pattern on the bottom:

Simplifying...
A%5BL%5D%2FA%5BS%5D+=+%2849%2B56sqrt%283%29%2B48%29%2F%2849-48%29
A%5BL%5D%2FA%5BS%5D+=+%2897%2B56sqrt%283%29%29%2F1
A%5BL%5D%2FA%5BS%5D+=+97%2B56sqrt%283%29