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Q.

A wire of $5024 m$ length is in the form of a square. It is cut and made to a circle. Find the ratio of the area of the square to that of the circle.

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a

\frac{11}{14}

b

\frac{14}{11}

c

\frac{51}{48}

d

\frac{1}{2}

answer is A.

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Detailed Solution

It is given that a wire of

5024 m

length is in the form of a square. It is cut and made to a circle.
We need to find the ratio of the area of the square to that of the circle.
By data,

Length of wire = 5024 m

The wire is in the form of a square.
Therefore, length of the wire is same as the perimeter of the square.
We know that the perimeter of a square with side

a

is

4 a

units.
Thus,

4 a = 5024 m

\Rightarrow a = \frac{5024}{4} m

We know that the area of a square with side

a

is

a^{2}

uni t^{2}

.
Thus,

Area of the square = a^{2}

\Rightarrow Area of the square = {(\frac{5024}{4})}^{2} m^{2}

— (1)
The wire is cut and made to a circle. Therefore, the circumference of the circle and the length of the wire will be equal.
We know that the circumference of a circle with radius

r

is

2 πr

units.
Therefore,

2 πr = 5024

\Rightarrow r = \frac{5024}{2 π}

We know that the area of a circle with radius

r

is

π r^{2}

uni t^{2}

.
Therefore,

Area of the circle = π r^{2}

\Rightarrow Area of the circle = π {(\frac{5024}{2 π})}^{2} m^{2}

— (2)
From eq(1) and eq(2),

\frac{Area of the square}{Area of the circle} = \frac{{(\frac{5024}{4})}^{2}}{π {(\frac{5024}{2 π})}^{2}}

\Rightarrow \frac{Area of the square}{Area of the circle} = \frac{(5024)^{2} \times 4 \times π}{(4)^{2} \times (5024)^{2}}

\Rightarrow \frac{Area of the square}{Area of the circle} = \frac{π}{4}

We know that,

π = \frac{22}{7}

\Rightarrow \frac{Area of the square}{Area of the circle} = \frac{22}{7 \times 4}

\Rightarrow \frac{Area of the square}{Area of the circle} = \frac{11}{14}

Thus the ratio of the area of the square to that of the circle is

\frac{11}{14}

.
Hence, the correct option is 1.

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