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Question

A 10 cm long rod AB moves with its ends on two mutually perpendicular straight lines OX and OY. If the end A be moving at the rate of 2 cm/sec, then when the distance of A from O is 8cm, the rate at which the end B is moving, is


A

83cm / sec

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B

43cm / sec

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C

29cm / sec

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D

None of these

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Solution

The correct option is A

83cm / sec


Explanation for the correct option:

Step1. Draw the diagram:

Here it is given that the rod AB moves with its ends on two mutually perpendicular straight lines OX and OY

Let the end A of rod move on line OX with respect to time t and end B of rod move on line OY with respect to time t .

Let the distance from point A to point O be x and the distance from point B to point O be y.

Step2. To find the rate at which the end B is moving i.e. dydt:

From the figure, we know that ΔBOAΔBOA is a right-angled triangle.

Therefore, from Pythagoras theorem we get,

x2+y2=100.....(1)

Differentiating on both side we will get,

2xdxdt+2ydydt=0.....(2)

It is given that end A is moving at the rate of 2 cms i.e. dxdt=2 cm

and the distance of A from O is 8cm i.e. x=8cm

Substituting the value of x in equation )(1) , we get 82+y2=102

y2=10064

y=36
So, y=6 cm

Hence the distance from point B to point O is 6cm.

From equation (1) and (2),

dydt=-166=-83

Thus, the rate at which B is moving is 83cm / sec

Hence, Option(A) is the correct answer.


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