SOLUTION: If an arc of 45 degrees on circle A has the same length as an arc of 30 degrees on circle B, find the ratio of the area of circle A to the area of circle B.

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Question 255263: If an arc of 45 degrees on circle A has the same length as an arc of 30 degrees on circle B, find the ratio of the area of circle A to the area of circle B.
Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!
The length of an arc is, in general:

degrees in the arc ------------------ * 2 * pi * r 360

If represents the radius of circle A then the arc length is:

which simplifies to

If represents the radius of circle B then the arc length is:

which simplifies to

We are told that these arc lengths are equal so

We are asked to find the ratio of the areas of the two circles. The area for circle A would be and for circle B it would be . The ratio then would be:

The pi's cancel leaving:

which can be rewritten as

So if we can find the ratio of the radii then we can find the ratio of the areas. Our arc length equation tells us that

We can use this to find the ratio of the radii. Multiplying both sides by 4 we get:

which simplifies to:

Dividing both sides by pi we get:

Dividing both sides by we get:

So the ratio of the radii is 2/3. We can put this into our area ratio expression:

and we get:

which simplifies to:

Note that we were able to find the ratios of the radii and the areas without being able to find any radius or any area!?