# Really hard related reates problem

## The Attempt at a Solution

V=20 S=6 E=2 H=20/12 (from V=SEH?)

$\frac{dS}{dt}=-.2$ <--- + OR -?

$\frac{dE}{dt}=.3$ <--- + OR -?

V=SEH

Product rule for 3 variables:
$\frac{dV}{dt}=SE\frac{dH}{dt}+EH\frac{dS}{dt}+SH \frac{dE}{dt}$

$0=SE\frac{dH}{dt}+EH(-.2)+SH(.3)$

$-EH(-.2)-SH(.3)=SE\frac{dH}{dt}$

$\frac{-EH(-.2)-SH(.3)}{SE}=\frac{dH}{dt}$

$\frac{-(2)(\frac{20}{12})(-.2)-(6)(\frac{20}{12})(.3)}{SE}=\frac{dH}{dt}$

$\frac{-7}{36}=\frac{dH}{dt}$

Negative, therefore decreasing.

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Dick
Homework Helper
The method of solution is fine. But picture the problem a little more carefully. It tells you the side walls are moving together at .2 m/s. This doesn't tell you dS/dt=(-.2). What does it tell you?

It means that they're getting smaller, therefore negative right? so that's why dS/dt is negative and not dE/dt?

Dick
Homework Helper
Does it mean that dS/dt=-.4? Not sure?

If the side walls are getting closer it's the end walls that are getting shorter, right?

If the side walls are getting closer it's the end walls that are getting shorter, right?

OH I see, one sec let me retry this.

Ended up getting -1/12 = dH/dt is this correct? So therefore the depth of the liquid is decreasing.

Dick
Homework Helper
Ended up getting -1/12 = dH/dt is this correct? So therefore the depth of the liquid is decreasing.

Well, I got +1/12=dH/dt. One of us made a mistake.

Um I think I made the mistake because I said dS/dt is negative, should be positive since the sides are getting bigger and the ends are getting smaller... But I'm not positive..

Dick
Homework Helper
Um I think I made the mistake because I said dS/dt is negative, should be positive since the sides are getting bigger and the ends are getting smaller... But I'm not positive..

What did you use for dS/dt and dE/dt?

dS/dt=-.3
dE/dt=+.2

Dick
Homework Helper
dS/dt=-.3
dE/dt=+.2

Now that's not right, is it? If the side walls are getting closer the end walls are getting shorter. So dE/dt should be negative.

That's what I thought I did wrong, thanks for all your help.

Much appreciated.