1. A circle with centre O has radius r cm. A sector of the circle, which has an angle of Ɵ radians at O, has perimeter 6 cm.
show that Ɵ = 6/r - 2 , and express the area A cm^2 of the sector in terms of r.
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You need to use the equation that gives the perimeter of sector of circle of `x^o` , such that:
`p = 2r + (x^o*pi*r/180^o)`
You need to convert the radians into degrees, such that:
`x^o = (theta*180^o)/pi`
Replacing `(theta*180^o)/pi` for `x^o` yields:
`p = 2r + (theta*180^o*pi*r)/((pi*180^o))`
Reducing duplicate factors yields:
`p = 2r + theta*r`
Since the problem provides the information that perimeter `p = 6` , yields:
`6 = 2r + theta*r => r*theta = 6 - 2r => theta = (6 - 2r)/r => theta = 6/r - (2r)/r => theta = 6/r - 2`
Hence, testing if the equation `theta = 6/r - 2` holds, under the given conditions, yields that the equation `theta = 6/r - 2` is valid.
You need to evaluate the area of circle sector using the following formula, such that:
`A = (theta*r^2)/2`
Replacing `(6 - 2r)/r` for `theta` yields:
`A = ((6 - 2r)/r*r^2)/2 => A = (6r)/2 - (2r^2)/2`
`A = 3r - r^2`
Hence, evaluating the area of circle sector, in terms of r, yields `A = 3r - r^2.`
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