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Q. 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

Application of Derivatives

Solution:

Let $AB = x\, m$,
$BC = y\, m$ and $AC = 10\, m$
$\therefore x^{2 }+ y^{2}= 100 \quad...\left(i\right)$
On differentiating $\left(i\right)$ w.r.t. $t$, we get
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$2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0$
$\Rightarrow 2x\left(3\right) - 2y\left(4\right)= 0$
$\Rightarrow x = \frac{4y}{3}$
On putting this value in $\left(i\right)$, we get
$\frac{16}{9} y^{2} + y^{2} = 100$
$\Rightarrow y^{2} = \frac{100 \times 9}{25} = 36$
$\Rightarrow y = 6\,m$