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Question
If the circumference of two circles are in the ratio 2 : 3, what is the ratio of their areas?
Solution
We are given ratio of circumferences of two circles. If `C=2 pir` and `C'=2pi r'` are circumferences of two circles such that
`C/C'=2/3`
⇒` (2pi r)/(2pi r')=2/3` ..............(1)
Simplifying equation (1) we get,
`r/(r')=2/3`
Let `A=pir^2` and `A'=pir^('2)` are the areas of the respective circles and we are asked to find their ratio.
`A/A'=(pir^2)/(pir^('2))`
`A/A'=r^2/r^('2)`
`A/A'=(r/(r'))^2` ...................(2)
We know that `r/(r')=2/3`substituting this value in equation (2) we get,
`A/(A')=(2/3)^2`
`⇒ A/(A')=4/9`
Therefore, ratio of their areas is `4:9`
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