An elastic belt is placed round the rim of a pulled of radius 5cm. One point on the belt is pulled directly away from the centre `O` of the pulley until it is at P, 10cm from O. Find the length of the belt that is in contact with the rim of the pulley. Also, find the shaded area.

Explanation

Let `/_AOP=/_BOP=theta`

In `/_\AOP`

`cos theta = (OA)/(OP)=5/(10)= 1/2`

Arc AB =`( 120^0 xx 2 xx pi xx 5)/(360^0) = (10xxpi)/3 cm `

Length of the belt that is in contact with the rim of the pulley
= Circumference of the rim -Length of arc AB

Length of the belt that is in contact with the rim of the pulley =` 2 xx pi xx 5 - (10xxpi)/3 = (20 xx pi )/3 cm `

Area of sector OACB= `(120^0 xx pi xx 5^2)/(360^0) = (25 xx pi)/3 `

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