Question:
The areas of two circles are in the ratio 4: 9. What is the ratio between their circumferences?
Solution:
Let the radii of the two circles be r and R, the circumferences of the circles be c and C and the areas of the two circles be a and A.
Now,
$\frac{a}{A}=\frac{4}{9}$
$\Rightarrow \frac{\pi r^{2}}{\pi R^{2}}=\left(\frac{2}{3}\right)^{2}$
$\Rightarrow \frac{r}{R}=\frac{2}{3}$
Now, the ratio between their circumferences is given by
$\frac{c}{C}=\frac{2 \pi r}{2 \pi R}$
$=\frac{r}{R}$
$=\frac{2}{3}$
Hence, the ratio between their circumferences is 2 : 3.
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