Answer:
-0.69 feet per second
Step-by-step explanation:
We have dx/dt = 2 ft/sec, and we want to determine dy/dt.
x and y are related by the Pythagorean Theorem.
x²+y²=16²
Differentiate both sides of this equation with respect to t to get
[TeX]2x\frac{dx}{dt}+2y\frac{dy}{dt} =0[/TeX]
[TeX]2y\frac{dy}{dt} =-2x\frac{dx}{dt}[/TeX]
Divide both sides by 2y
[TeX]\frac{dy}{dt} =-\frac{x}{y}\frac{dx}{dt}[/TeX]
When x = 5 ft, we have
x²+y²=16²
5²+y²=256
y²=256-25=231
therefore
y= [TeX] \sqrt{213} [/TeX]
[TeX]\frac{dy}{dt} =-\frac{5}{\sqrt{213}}\frac{2 ft}{sec}[/TeX]
=-0.69
The top of the ladder is sliding down (because of the negative sign in the result) at a rate of -0.69 feet per second.
