I was doing this simple Calc 1 problem and it took me forever to get it right and it was embarrassing. I could see that the problem was easy but I just couldn't 'see' what I was doing. I couldn't find the path. I solved it, and then I tried to figure out how I had done it and why it took so long. The problem was this:
There is a circular cone-shaped tank. It's pointed upward. It's filling at a rate of $12 m^3/s$. The radius of the base is $26m$. The height of the tank is $8m$. The water's surface is circular and has radius $r$. The height of the water is $h$. What is $\frac {\delta h}{\delta t}$ when $r=10$?
I know that $v=\frac{\pi r^2h}{3}$. I also knew that $\frac{26}{8}=\frac{r}{h}$.
After I solved it, I came up with a 'map' for the work I had done. This is what I did:
given: $v', r$
know: {r,h} {v,r,h}
need: {$h'$, {r,h}}
The know line reads: I know a function connecting r and h. I know a function connecting v,r and h.
The need line reads: I need a function connecting $h'$ and r. r is in bold because I can use the {r,h} function that I know, and the r value I was given.
The process was then:
{v,r,h} start with volume function.
{v,{r:h}, h} replace r with h using my known {r,h} function.
{v', h', h} differentiate to get these variables. v' I have, h' I want, h I don't need.
{v', h', {r,h}} replace h with r using known {r,h} function.
Now the problem is solved: I had v', I needed h', and I had r.
SO, my question is this: is there a system somewhere comparable to this process? Does anyone else do this?