how to find the perimeter of the shaded region in terms of π (pi) ?
The diagram shows a quadrant with centre O and of radius, 9 cm. Given that the arc AM is 6 cm and the angleAOM is 2/3 rad , how to find the perimeter of the shaded region in terms of π (pi) ?
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You need to write the formula of perimeter of shaded region to check what the problem provides and what you need to find, such that:
`P = MO+OB+MB`
Notice that `MO = OB = R = 9` and you need to find the length of the arc MB.
You should remember that you may find the length of arc using the following formula such that:
`MB = pi*R*(hat(MOB))/180`
`hat(MOB) = pi/2 - 2pi/3 =gt hat(MOB) = (3pi-4pi)/6=gt hat(MOB) = -pi/6 =gt MB = -9pi/6`
Substituting -`9pi/6` for MB in equation of perimeter yields:
`P = 9 + 9- 9pi/6 `
`P = 9(2- pi/6) =gt P = (9/6)(12 - pi) =gt P = (3/2)(12 - pi)`
Hence,evaluating the perimeter in terms of `pi` , using the given information, yields `P = (3/2)(12 - pi).`
Required perimeter P= MO+OB+BM
angle AOB = pi/2 radian being angle of the quadrant
angle AOM = 2/3 radian
therefore angle MOB = pi/2-2/3
length of arc MB = angle*radius = (pi/2-2/3)*9 = 9pi/2-6 cm
P = 9+9+9pi/2-6 = 12+9pi/2 = 26.1cm appoximately
The perimeter of the shaded portion is 12+9pi/2 cm or 26.1cm approximately
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