# Determine how fast the top of the ladder is decreasing when the foot of the ladder is 3 meters from the house. A 5 meter long ladder is leaning against the side of a house. the base of the ladder...

Determine how fast the top of the ladder is decreasing when the foot of the ladder is 3 meters from the house.

A 5 meter long ladder is leaning against the side of a house. the base of the ladder is pulled away from the house at a rate of .4m/sec. Determine how fast the top of the ladder is decreasing when the foot of the ladder is 3 meters from the house. List the unknown,constant and given

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Let the base of the wall be at the origin, the distance from the foot of the wall be `x` , and the height where the ladder meets the wall be `y` .

Since the ladder is 5m long, the pythagorean theorem gives us :

`x^2+y^2=25` . We are given `x=3m,(dx)/(dt)=.4m/sec` and we are wanting to compute `(dy)/(dt)` .

Differentiate with respect to `t` :

`2x (dx)/(dt)+2y(dy)/(dt)=0` . Now `x=3 =>y=4` and `(dx)/(dt)=.4` ;substituting these values gives us:

`2(3)(.4)+2(4)(dy)/(dt)=0`

`8(dy)/(dt)=-2.4`

`(dy)/(dt)=-.3`

**So when the base of the ladder is 3m from the wall, the top of the ladder is sliding down the wall at a rate of .3 m/sec.**