This interactive model is based on related rates ladder problems and uses the Pythagorean Theorem to maintain the length of the ladder as one of its anchors moves.
I gave the ladder a length of 8ft, so, if the top of the ladder is \(y\) ft from the ground and the bottom is \(x\) ft away from the wall, the ladder is defined by \(x^2+y^2=64\). There's also a button to make it start sliding down at \(2\frac{ft}s\).
If we know the position and speed of one point, we can see how fast the other point is sliding by deriving with respect to time, which looks like:
\(\frac d{dt}[x^2 + y^2=64]\)
\(\frac d{dt}[x^2] + \frac d{dt}[y^2]=0\)
\(2x\frac{dx}{dt} + 2y\frac{dy}{dt}=0\)
Let's see what how fast the bottom of the ladder is sliding when the ladder is 4ft from the top of this 10ft tall structure:
\(x=\sqrt{(8)^2-(10-4)^2}=\sqrt{64-36}=\sqrt{28}=2\sqrt{7}\)
\(2(2\sqrt7)\frac{dx}{dt}+2(6)(-2)=0\)
\(4\sqrt7\frac{dx}{dt}-24=0\)
\(\frac{dx}{dt}=\dfrac{24}{4\sqrt7}=\dfrac{6}{\sqrt7}\)
\(\frac{dx}{dt}\) is positive, so the distance between the bottom of the ladder is increasing at a rate of \(\dfrac{6}{\sqrt7}\frac{ft}s\).