How fast is the top of a 13-foot ladder sliding down a wall at the instant the base is 12 feet from the wall, assuming the base is moving away from the wall at 4 feet per second?

To determine how fast the top of the ladder is sliding down the wall, we can apply the Pythagorean theorem and related rates of change.

Given that the ladder forms a right triangle with the wall and the ground, we can represent the lengths using variables. Let ( y ) be the height of the top of the ladder on the wall, and ( x ) be the distance from the wall to the base of the ladder on the ground. According to the Pythagorean theorem, we have:

[

x^2 + y^2 = L^2

]

Where ( L ) is the length of the ladder. In this case, ( L = 13 ) feet. We know that ( x = 12 ) feet and that the base is moving away from the wall at a rate of ( \frac{dx}{dt} = 4 ) feet per second.

First, we can find ( y ) by plugging in our value for ( x ):

[

12^2 + y^2 = 13^2 \

144 + y^2 = 169 \

y^2 = 169 - 144 \

y^2 = 25