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In figure, a square OABC is inscribed in a quadrant O P B Q . If $O A = 15 cm,$ find the area of the shaded region.

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120.6 c m^{2}

None of these

150 c m^{2}

128.25 c m^{2}

answer is A.

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Detailed Solution

Given that, the side of the square is represented by 15 cm.
Question Image

The difference between the area of quadrant and area of square is equal to the shaded region.
The square is representing two triangle which are VOAB and VOAB .
Determining the hypotenuse of the triangle which is OB by using the Pythagorean theorem

H 2 = B 2 + P 2

H 2 = 152 + 152

\Rightarrow

H 2 = 450

\Rightarrow

H = 21.21
The length of OB is represented by 21.21 cm and also represents the radius of the circle.
Determining the area of quadrant

A_{1}

by using the formula

A 1 = q 360 π r 2

in which q represents the angle which is formed from origin of the circle to its surface that is line OB.

A 1 = 90360 3.14 21.21 2

⇒ 14 × 3.14 × 450

\Rightarrow

353.24

{cm}^{2}

Determining the area of the square

A_{2}

by using the form

A_{2} = (Side)^{2}

A 2 = 152

\Rightarrow

225

{cm}^{2}

Determining the shaded region S by subtracting the area of square

A_{2}

from the area of quadrant

A_{!}

,
S = 353.25

-

225

\Rightarrow

128.25

{cm}^{2}

The area represents the shaded region is 128.25

{cm}^{2}

.
Hence the correct option is 1.

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