Question

A ladder 7.5 m long leans against a wall. The ladder slides along the floor away fiom the Wall at the rate of 3 cm/sec. How fast is the height of the ladder on the wall decreasing, when the foot of the wall is 6 m away from the wall?

Answer 1

Answer:

4 cm/sec

Step-by-step explanation:

In the question,

Length of the ladder, l = 7.5 m = 750 cm

Rate of the sliding away of ladder from the wall, db/dt = 3 cm/sec

Distance of the foot of the ladder from the wall = 6 m = 600 cm

So,

At that time using Pythagoras theorem in the right triangle, we get,

$l^{2} = b^{2} + h^{2} \\7.5^{2} = 6^{2} + h^{2} \\56.25 = 36 + h^{2} \\h=\sqrt{20.25} =4.5m$

h = 450 cm (at that time)

Now,

Also,

$l^{2}= b^{2}+ h^{2}$

On differentiating it w.r.t time we get,

$0=2b\frac{db}{dt} +2h\frac{dh}{dt} \\b\frac{db}{dt} =-h\frac{dh}{dt}\\600(3)=-450(\frac{dh}{dt} )\\\frac{dh}{dt} =-4cm/sec$

(because, length of the ladder, l is constant. )

Therefore, the rate of decreasing of height of the ladder = 4 cm/sec.