Not related to the Umbrella Corporation
If the area of the shaded region equals
7
x
−
cos
x
, what is the value of
x
π
?
The answer is 6.
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3 solutions
Yay! Correct! Doesn't this look like an umbrella?
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Yes, it does and it reminded me of the recent Hong Kong Umbrella Revolution. Sorry to be political.
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I solved it using integration.........
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@Rajat Kharbanda – Yes, post solution. I am not very fond of calculus and integration.
Lol! Umbrella Corp . haha
truly a fantastic solution
Sir , how did u get that the 2 parts of circle with radius 1cm occupies 1/3 area of the original circle i.e. third line of ur answer and also that part of circle that occupies 1/6 area of another circle(i.e. fourth line) ??
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I have changed the figure. 3 π and 6 π indicate the one-third and one-sixth of a radius-1 circle respectively. Hope it helps.
Total required area = 2*Pink area=All area -2( green+blue+red) areas.
R
e
q
u
i
r
e
d
a
r
e
a
=
2
∗
π
−
2
∗
(
2
1
∗
6
π
−
4
3
−
2
1
∗
3
2
∗
π
)
=
{
6
7
∗
π
−
2
3
}
=
6
7
∗
π
−
C
o
s
6
π
=
X
−
C
o
s
X
⟹
6
π
=
X
.
.
S
o
X
π
=
6
.
The area of the shaded region
A = Area of the radius-2 semicircle − 2 × Area of 3 1 of radius-1 circle − Area of 6 1 of radius-1 circle − 2 × Area of side length-1 equilateral triangle = 2 1 π ( 2 2 ) − 2 × 3 1 π − 6 1 π − 2 × 2 1 ( 1 ) ( 2 3 ) = ( 2 − 3 2 − 6 1 ) π − 2 3 = 6 7 π − 2 3 = 6 7 π − cos ( 6 π )
⇒ x = 6 π ⇒ x π = 6