The emergent velocity, v, of a liquid flowing from a hole in the bottom of a tank is given by v = Square root(2gh) where g is the acceleration due to gravity and h is the depth of the liquid in the tank. Find the rate of change of v with respect to h when (a) h = 9 , (b) h = 4 .

Question

mathmate · Answer

$ v=\sqrt{2gh} $
can you find $ \large \frac{dv}{dh} $ ?

anonymous · Answer

Would it be $$\frac{ 1 }{ 2 }(2gh)^{-\frac{ 1 }{ 2 }}$$ ?

mathmate · Answer

You can take the constant $ \sqrt{2g} $ out before you integrate.

mathmate · Answer

*differentiate.

anonymous · Answer

And have it as $$\sqrt{2g} + \sqrt{h} $$ ?

mathmate · Answer

"Take out" is meant to factorize and put it on the left of the differentiation:
 $ \frac{d\sqrt{2gh}}{dh}  \
=\sqrt{2g}\frac{d\sqrt{h}}{dh} \
=\sqrt{2g} {\frac{1}{2}}\frac{1}{\sqrt{h}} \
=\frac{\sqrt{g}}{2h}
$

anonymous · Answer

Ah, that is far more elegant. lol

anonymous · Answer

Do I just input the values now?

mathmate · Answer

No, there is a typo.  it should read
$ \frac{\sqrt{2g}}{2\sqrt{h}} $
Typo is all on the last line.

anonymous · Answer

That makes much more sense.

mathmate · Answer

Do check answers and be convinced that they are correct before you put it into your answer, i.e.  before you "own" it.

anonymous · Answer

I usually do. I was sure I was missing something and have been thinking it though the entire time. I thought you had pulled out the square root of two and somehow turned it into two on the bottom. It made my brain fart. ><;

mathmate · Answer

That's good.  Getting the right answer is a temporary pleasure.  Knowing how to solve the problem is the real success!
Keep up the good work!

anonymous · Answer

So, since that is the first derivative, that represent rate of change, so plug in the force of gravity (32 ft/sec) and the two h's given, right? (Yeah! Or in Isaac Asimov's words: "The true pleasure is in the finding out, rather than the knowing." :D

mathmate · Answer

Exactly!
Note: are you working in imperial system or metric?  32.2 ft/s^2 or 9.81 m/s^2 ?

mathmate · Answer

I agree with that too!

anonymous · Answer

Imperial. Mayhaps someday the United States will convert to metric. >.>