A ladder 13 m long is leaning against a wall. The bottom of the ladder is pulled along the ground away from the wall at the rate of 4 m/sec. Find the rate of decreasing at which the top of the ladder moving downwards on wall when the foot of the ladder is 5 m away from the wall.

Options

A. `(-5)/3 m//sec.`
B. `(5)/3 m//sec.`
C. `(-10)/3 m//sec.`
D. `(10)/3 m//sec.`

Answer

A

Explanation

Let at any time 't' , the foot of the ladder be 'x' m away from the wall and the top of the ladder is at 'h' m height from the ground.
`:. h^(2)+x^(2)=13^(2)`
`rArr h = sqrt(169-x^(2))`
`rArr (dh)/(dt)=(-2x)/(2sqrt(169-x^(2))).(dx)/(dt)`
` At x = 5 m, (dx)/(dt)=4 m//sec`
` :. " "(dh)/(dt)= (-5)/(sqrt(169-5^(2)))xx4`
`=(-5 xx 4 )/(12) = (-5)/3 m//sec.`
Therefore the top of the ladder is moving downwards at a rate of `5/3 m//sec.`