[tex]-5.3\:\text{ft/s}[/tex]
Step-by-step explanation:
We start by applying the Pythagorean theorem to the ladder, with its length L as the hypotenuse:
[tex]L^2 = 100\:\text{ft}^2 = x^2 + y^2[/tex]
where x is the vertical distance from the top of the ladder to the ground and y is the horizontal distance from the bottom of the ladder to the wall. Taking the derivative of the above expression with respect to time, we get
[tex]0 = 2x\dfrac{dx}{dt} + 2y\dfrac{dy}{dt}[/tex]
Solving for dx/dt, we get
[tex]\dfrac{dx}{dt} = -\left(\dfrac{y}{x}\right)\dfrac{dy}{dt} = -\left(\dfrac{\sqrt{L^2 - x^2}}{x}\right)\dfrac{dy}{dt}[/tex]
We know that
[tex]\dfrac{dy}{dt} = 4\:\text{ft/s}[/tex]
when x = 6 ft. So the rate at which the top of the ladder is going down is
[tex]\dfrac{dx}{dt} = -\left(\dfrac{\sqrt{100\:\text{ft}^2 - (6\:\text{ft})^2}}{6\:\text{ft}}\right)(4\:\text{ft/s})[/tex]
[tex]\:\:\:\:\:\:\:= -5.3\:\text{ft/s}[/tex]
The negative sign means that the distance x is decreasing as y is increasing.
