Related Rates: 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...

Related Rates: 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?

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should draw a right angle triangle whose hypotenuse measures 16 feet.

You should come up with the notations for the legs of triangle such that: x-horizontal leg; y-vertical leg.

You should use Pythagora's theorem such that:

`x^2 + y^2 = 16^2 =gt x^2 + y^2 = 256`

You need to differentiate both sides with respect to t such that:

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

You need to factor out 2 to the left such that:

`2(x(dx)/(dt) + y(dy)/(dt)) = 0 =gt (x(dx)/(dt) + y(dy)/(dt)) = 0`

The problem provides the information that the base of ladder  moves away from the bottom of the building at a rate of 2 feet per second, hence, you need to substitute -2 for `(dx)/(dt)`  and 3 for  `(dy)/(dt)`  such that:

`-2x + 3y = 0`

You should write y in terms of x, using `x^2 + y^2 = 256`  such that:

`y = sqrt(256 - 9) =7gt y = sqrt24`

`-2x + 3sqrt247 = 0 =gt x = 3sqrt247/2`

Hence, evaluating how fast is Fabio falling yields `3sqrt247/2`  per second.