question_answer

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)  \[4\sqrt{3}m\]                 

B)  \[5\sqrt{3}m\]

C)  \[5\sqrt{2}m\] 

D)         \[8\,m\]

E)  \[6\,m\]

Correct Answer: E

Solution :

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