Question

# A ladder 10 m long rests against a vertical wall with the lower end on the horizontal ground. The lower end of the ladder is pulled along the ground away from the wall at the rate of $$3 cm/s$$ The height of the upper end while it is descending at the rate of $$4 cm/s$$, is

A
43m
B
53m
C
52m
D
6m

Solution

## The correct option is D $$6m$$Let $$AB=x, BC=y$$ and $$AC=10m$$$$\therefore x^2+y^2=100$$ ....... $$(i)$$$$\Rightarrow 2x\dfrac{dx}{dt} + 2y\dfrac{dy}{dt}=0$$$$\Rightarrow 2x(3)-2y(4)=0$$          $$\left[\because \dfrac{dx}{dt}=3cm/s, \dfrac{dy}{dt} = -4cm/s\right]$$$$\Rightarrow 6x-8y=0$$    $$\Rightarrow x=\dfrac{4y}{3}$$On putting this value in Eq. $$(i)$$, we get$$\dfrac{16}{9}y^2+y^2=100$$$$\Rightarrow \dfrac{25y^{2}}{9}=100$$$$\Rightarrow y^{2}=36$$$$\Rightarrow y = 6$$Hence, the height of the upper end is $$6m$$.Mathematics

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