A sector is cut from a circular sheet of radius 100 cm, the angle of the sector being 240°. If another circle of the area same as the sector is formed, the radius of the circle of the new radius is.

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79.5 cm

81.3 cm

83.4 cm

88.5 cm

answer is B.

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Detailed Solution

The diagram of the conditions given in the question is as follows:

We see that the center of the circle is O and the radii are AO and BO. The length of the radius is 100 cm.
The main sector OAB is said to be cut out of the circle.
The cutting angle is 240°.
Now we find the area of the sector.
We know that the area of a circle is given as

A c = π r 2

, r is the radius of the circle.
We know that the angle of a circle is 360° or 2π radians.
So, for

2 π

radians, the area is

π r 2

.
Then, on comparing, area for θ will be given as

2 π θ = π r 2 A s

Hence, area of the sector is given as

A s = 12 r 2 θ

, where θ is the angle of sector in radians.
We know that the angle of the sector in our question is 240°.
We shall convert this into radians. We know that 180° is equal to π. Hence, 240° will be given as

180240 = π θ

Therefore,

θ = 4 π 3

radians
We will substitute

θ = 4 π 3

and r = 100 in the formula for area of sector

A s = 12 r 2 θ ⇒ A s = 12 (100) 2 43 π ⇒ A s = 100002 43 π

Now given that a circle of equal area is created. Let the radius of the circle be r.
We compare the area of the circle with the radius r and the area of the sector and find the value of r.