A ladders 5 ~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 2 ~m / sec . The speed at which its height on the wall decreases when the foot of the ladder is 4 ~m away from the wall is

Step-by-step solution

Let A B be the ladder of length 5 ~m . We are given, d x d t =2 ~m / sec In A B C aligned & A B²=A C²+B C² &(5)²=x²+h² ...(i) aligned Differentiating Eq. (i) w.r.t. t , we get aligned 0 &=2 x d x d t +2 h d h d t d h d t &= -x h d x d t ...(ii) aligned Now, from Eq. (i), when x=4 h²=25-16 h²=9 h=3 ~m From Eq. (ii), we get [ d h d t ]= -4 3 2= -8 3 The negative sign shows the height decreases and decreasing rate is 8 3 ~m / sec .