Question

Consider a circle with unit radius .There are seven adjacent sectors S₁, S₂ , S₃ , .....S₇ in the circle such that their total area is 1/8 of the area of the circle.Further the area of the 'j'th sector is twice that of (j - 1) sector, for j = 2,....,7 what is the angle in radians,subtended by the arc of S₁, at the center of the circle.

A) π/508
B) π/2040
C) π/1016
D) π/1524

Explain with appropriate calculations.

Answer 1

S1 , S2 , S3 , ...........S7 are the adjacent sectors in the circle of radius 1 unit .

let ∅1 , ∅2 , ∅3 .........∅7 are the angle subtend by S1 , S2, S3 .......S7 arc respectively .

area of S1 sector = ∅1/360° × πr²
area of S2 sector = ∅2/360° × πr²
................
.....
area of S7 sector = ∅7/360× πr²

question ask ,
area of j th sector = 2× area of (j -1)th sector .
where j =2, 3, .......7

it means ,

area of S2 sector = 2× area of S1 sector
∅1/360× πr² = 2∅/360 × πr²
∅2 = 2∅1

similarly ,
∅3=2∅2
∅4 =2∅3
∅5 =2∅4
∅6 =2∅5
∅7 =2∅6

if we solve this
we find ,
∅2 = 2∅1
∅3 = 4∅1
∅4 =8∅1
∅5 =16∅1
∅6=32∅1
∅7 =64∅1

again question, ask

area of total sectors = 1/8 area of circle
(∅1 + ∅2 +.....∅7)/360° × πr² = 1/8 ×πr²

(∅1 + ∅2+.....∅7) = 360/8°

so,
(∅1 + 2∅1 +4∅1 +....64∅1) =360°/8 ×2π/360 = π/4

∅1( 1+ 2 +4 + 8 .....64) =π/4

1 + 2 + 4 + 8 + 16....64 are in GP
so,
64 = 1.(2)^(n-1)
n =7

Sn = 1(2^7 -1)/(2-1) = 2^7 -1 = 127

so,
∅1 × 127 =π/4

∅1 = π/127× 4 = π/508

hence , arc S1 subtended π/508 rad

option (A ) is correct .

Answer 2

by using some property of sectors...we can solve this....

answer is option (A).

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hope it will help ...

option C can't be answer of this question...I checked many times...I m 100% sure...option (A) is correct