Optimizing V=SEH: Solving Really Hard Related Rates Problem | dH/dt = -7/36

[iRaid]

Homework Statement

Homework Equations

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.

 

[Dick]

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?

 

[iRaid]

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

 

[Dick]

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?

 

[iRaid]

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.

 

[iRaid]

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

 

[Dick]

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.

 

[iRaid]

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]

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?

 

[iRaid]

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

 

[Dick]

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.

 

[iRaid]

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

Much appreciated.