Question

A 13 foot ladder is leaning against the wall of a house. the base of the ladder slides away from the wall at a rate of 0.75ft/sec. how fast is the top of the ladder moving down the wall when the base is 12 feet from the wall.

I figured out its a 13, 12, 5 triangle but thats about all i have

Answer 1

You're right that when \(x=12\) you have \(y=5\). You probably found that using the Pythagorean theorem. Well bring up the equation again, this time in terms of unknown side lengths:
\[x^2+y^2=13^2\]
We want to find \(\dfrac{dy}{dt}\) with all this given information. Do some implicit differentiation (with respect to \(t\)):
\[\begin{align*}\frac{d}{dt}\left[x^2+y^2\right]&=\frac{d}{dt}13^2\\
\frac{d}{dt}\left[x^2\right]+\frac{d}{dt}\left[y^2\right]&=0\\
2x\frac{dx}{dt}+2y\frac{dy}{dt}&=0
\end{align*}\]
Plug in the given infor and solve for \(\dfrac{dy}{dt}\):
\[2(12)(.75)+2(5)\frac{dy}{dt}=0\]