The sum of the radii of two circles is $84 c m$ and the difference of their areas is $5544$ $c m^{2}$ . Calculate the radii of the two circles.

52.5 c m, 31.5 c m

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52.5 c m, 30.5 c m

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50.5 c m, 31.5 c m

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50.5 c m, 30.5 c m

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Solution

The correct option is A $52.5 c m, 31.5 c m$
Let the two radii be $r_{1}$ and $r_{2}$
Given,
$r_{1} + r_{2} = 84$ and
$π {r^{2}}_{1} - π {r^{2}}_{2} = 5544$
Now,
$r_{1} + r_{2} = 84$
$=> r_{1} = 84 - r_{2}$ (i)
Again,
$π {r^{2}}_{1} - π {r^{2}}_{2} = 5544$
Using (i), we get,
$=> π (84 - r_{2})^{2} - π {r^{2}}_{2} = 5544$
$=> π [(84)^{2} - 2 (84) r_{2} + {r^{2}}_{2}] - π {r^{2}}_{2} = 5544$
$=> 7056 π - 168 r_{2} π - 5544 = 0$
$=> 42 π - r_{2} π - 33 = 0$
$=> 42 (3.14) - 3.14 r_{2} - 33 = 0$
$=> 98.88 - 3.14 r_{2} = 0$
$=> r_{2} = \frac{98.88}{3.14}$
$= 31.49$
$≃ 31.5$
Therefore,
Putting the value of $r_{2}$ in (i), we get,
$r_{1} = 84 - 31.5$
$= 52.5$
Thus,
the radii are $52.5$ and $31.5$

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