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a wire when bent in the form of a square encloses an area of 992.25 cm if the same wire is bent to form a semi circle, what will be the radius of the semi circle so formed? ​

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A wire when bent in the form of a square encloses an area of 992.25 cm2992.25 cm2 . If the same wire is bent in the form of a semicircle, what will be the radius of the semicircle so formed?

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Hint: To find the radius of the semicircle formed from a wire that is used to make a square, we will find the side of the square using the given area and equation Area=a2Area=a2 . Since the same wire is used to make a semicircle, we know that the perimeter of the square = circumference of the semicircle. Let us use the formula for circumference of semicircle, that is, πr+2rπr+2r and substitute this to the perimeter of the square to get the value of the radius.

Complete step by step answer:

We need to find the radius of the semicircle obtained. It is given that the wire used to create the semicircle was bent in the form of a square with an area 992.25 cm2992.25 cm2 .

Hence, we can write the area of square of side a as

Area=a2Area=a2

We know that the given area is 992.25 cm2992.25 cm2 .Hence,

a2=992.25a2=992.25

Now, let us take the square root to get the side of the square.

a=992.25−−−−−√=31.5cma=992.25=31.5cm

Let us find the perimeter of the square.

We know that perimeter of a square is given by

Perimeter=4aPerimeter=4a

Let us now substitute the values. We will get

Perimeter=4×31.5=126cmPerimeter=4×31.5=126cm

It is given that the same wire is used to form a semicircle. Hence,

Perimeter of the square = circumference of the semicircle…(i)Perimeter of the square = circumference of the semicircle…(i)

We know that the circumference of a semicircle with radius r is given as

Circumference=πr+2rCircumference=πr+2r

Let us substitute this in (i). We will get

πr+2r=126πr+2r=126

Let us take r outside from LHS. We will get

r(π+2)=126r(π+2)=126

We can take the value of π=227π=227 . Hence, the above equation becomes

r(227+2)=126r(227+2)=126

Let us solve us. We will get

r(367)=126r(367)=126

Taking the constants to one side, we will get

r=126×736r=126×736

We can now solve this. We will get

r=24.5cmr=24.5cm

Hence, the radius of the semicircle is 24.5 cm.