# How fast is Fabio falling at that instant?Fabio stands atop his 16 foot ladder when he realizes that the ladder is slipping down the side of the building. He decides that the base of the ladder is...

How fast is Fabio falling at that instant?

Fabio stands atop his 16 foot ladder when he realizes that the ladder
is slipping down the side of the building. He decides that the base of
the ladder is moving away from the bottom of the building at a rate
of 2 feet per second when it is 3 feet from the bottom of the building.
How fast is Fabio falling at that instant?

jeew-m | Certified Educator

The situation can be seen in this graph.

Lets take in an instance;

The distance from foot of building to the end of ladder at ground as x feet and foot of building to the top of ladder as y feet.

Since the ladder length is constant at 16ft we can write;

`x^2+y^2 = 16^2`

Now let us differentiate both sides with respect to time t;

`2x*dx/dt+2y*dy/dt = 0-----(1)`

dx/dt means the rate that which ladder slips in ground and dy/dt is the rate at which Fabio falling down.

It is given that x=3ft; dx/dt=2ft/s

So when x=3 then `y = sqrt(16^2-3^2) = 15.716`

Let us insert this data to (1)

`2*3*2+2*15.716*dy/dt`   = 0

`dy/dt` = -0.3817

= -0.38 ft/s

We have measured y is positive at the direction upward. But here y is changing at the downward direction. That is the reason for - sign.

The rate at which Fabio falling down at the instance of x=3 is 0.38 ft/s