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Q.

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


Question Image

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a

54.58 cm and 11.87 cm  

b

14.58 cm and 24.87 cm  

c

19.52 cm and 14.87 cm  

d

Question Imagecm and 17.08  cm 2   

answer is D.

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Detailed Solution

Given that the rim of a pulley of radius, Question Imagecm.
We have to find the length of the belt that is still in contact with the pulley, also the shaded area.
The required figure geometry is shown below,
Question ImageThe expression to find the area of a triangle is A= 1 2 b h  , where b and h are the base and height of a triangle.
Since, the tangent of a circle is perpendicular to the radius of the circle through the point of contact.
So,Question Imagecm.
Then, by Pythagoras theorem in △OAP  , we get,
Question ImageThe length of the belt that is still in contact with the pulley.  Thus,  cos θ 1 = OA OP ⇒cos θ 1 = 5 10 ⇒cos θ 1 = 1 2 ⇒ θ 1 = 60 °     
Since, ∠AOP=∠BOP  
Therefore,  ∠AOB= θ 2     
⇒ θ 2 =∠AOP+∠BOP   
⇒ θ 2 = 60 ° + 60 °    
⇒ θ 2 = 120 °  
Thus, the length of the belt that is still in contact with the pulley, L is, Question ImageThe area of the shaded region, A s   is, Question ImageThus, the length of the belt that is still in contact with the pulley and area of shaded region is Question Imagecm and 17.08  cm 2   respectively.
Therefore, the correct option is 4.
 
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