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The area of a circle is increased by $22 c m^{2}$ when its radius is increased by $1 cm$ . The original radius of the circle is,

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6 cm

3 cm

4 cm

3.5 cm

answer is B.

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Detailed Solution

It is given that the area of a circle is increased by

22 c m^{2}

when its radius is increased by

1 cm

. We need to find the original radius of the circle.
Let the original radius of the circle be

r

cm.
Then, the new radius of the circle will be

(r + 1)

cm.
We know that the area of a circle is

π r^{2}

sq.units, where

r

is the radius of the circle.
Therefore,
Original area of the circle is

π r^{2}

uni t^{2}

.
New area of the circle is

π {(r + 1)}^{2}

uni t^{2}

.
By data,
New area - Original area = 22

\Rightarrow π {(r + 1)}^{2} - π r^{2} = 22

\Rightarrow π (r^{2} + 2 r + 1) - π r^{2} = 22

[because

(a + b)^{2} = a^{2} + 2 ab + b^{2}

]

\Rightarrow π r^{2} + 2 πr + π - π r^{2} = 22

\Rightarrow 2 πr + π = 22

\Rightarrow π (2 r + 1) = 22

\Rightarrow 2 r + 1 = \frac{22}{π}

\Rightarrow 2 r + 1 = \frac{22 \times 7}{22}

[because

π = \frac{22}{7}

]

\Rightarrow 2 r + 1 = 7

\Rightarrow 2 r = 7 - 1

\Rightarrow 2 r = 6

\Rightarrow r = \frac{6}{2}

\Rightarrow r = 3

cm
If the area of a circle is increased by

22 c m^{2}

when its radius is increased by

1 cm

, then the original radius of the circle is

3

cm.
Hence, the correct option is 2.

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