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Question

A ladder, 5 metre long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides down wards at the rate of 10 cm/sec, then find the rate at which the angle between the floor and ladder is decreasing when lower end of ladder is 2 metres from the wall.


Solution

Length of the ladder=500cmLet the horizontal length covered between the wall and the ladder be x and vertical length covered between the wall and the ladder be y.And let the angle between the floor and ladder be θ.Then, sinθ=y500On differentiating with respect to t, we getcosθdθdt=1500dydt          ...(1)It is given that dydt=-10 cm/sec.        ...(2)Also,cosθ=x500When x=200 cm, cosθ=200500=25        ...(3)Substituting (2) and (3) in (1), we get25dθdt=1500-10dθdt=-120 radian/secondHence, the angle between the floor and the ladder is decreasing at the rate of 120 radian/second.

Mathematics
RD Sharma XII Vol 1 (2017)
Standard XII

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