May 12th, 2017, 06:45 AM  #1 
Newbie Joined: May 2017 From: alabama Posts: 6 Thanks: 0  calculus
A tank of water is being drained through an outlet, the height H (m) of water in the time t (s) is given by dh/dt = (3x10^3) √H given t = 0 and H = 4 determine an expression for H in terms of t. Can anybody point me in the right direction with this question? I don't really understand what it's asking. Last edited by skipjack; May 16th, 2017 at 06:02 AM. 
May 12th, 2017, 07:08 AM  #2  
Math Team Joined: Jan 2015 From: Alabama Posts: 2,731 Thanks: 707  Quote:
Quote:
Quote:
You can write this as $\displaystyle \frac{dH}{H^{1/2}}= H^{1/2}dH= (3x10^{3})dt$. Now integrate both sides. That will give you a "constant of integration". Use H(0)= 4 to determine that constant. Last edited by skipjack; May 16th, 2017 at 06:03 AM.  
May 12th, 2017, 10:43 AM  #3 
Member Joined: Feb 2015 From: Southwest Posts: 96 Thanks: 24 
I'm assuming here that the x is the symbol for multiplier here. In my example, I will use less confusing notation as x is a common variable. $\displaystyle \frac{dH}{dt}=(3\cdot 10^{3})\sqrt{H(t)}$ $\displaystyle \frac{dH}{\sqrt{H(t)}}=(3\cdot 10^{3})dt$ $\displaystyle \int H(t)^{\frac{1}{2}}dH=\int 3\cdot 10^{3}dt$ $\displaystyle 2\sqrt{H(t)}+C_1=3\cdot 10^{3}t+C_2$ $\displaystyle \sqrt{H(t)}=\frac{3\cdot 10^{3}}{2}t+C$ $\displaystyle H(t)=\bigg (\frac{3\cdot 10^{3}}{2}t+C\bigg )^2$ $\displaystyle H(0)=4$ $\displaystyle 4=\bigg (\frac{3\cdot 10^{3}}{2}(0)+C\bigg )^2$ $\displaystyle 2=C$ $\displaystyle H(t)=\bigg (\frac{3\cdot 10^{3}}{2}t+2\bigg )^2$ $\displaystyle H(t)=(.0015t+2)^2$ Hopefully, I didn't make a careless mistake. Hope this helps. Last edited by skipjack; May 16th, 2017 at 06:05 AM. 
May 16th, 2017, 05:06 AM  #4 
Newbie Joined: May 2017 From: alabama Posts: 6 Thanks: 0 
I think I may have noted the question down wrong, by 3x10^3 I meant as in the metrix prefix (0.003) rather than 3 multiplied by ten to the power of minus 3 if that makes sense?

May 16th, 2017, 05:56 AM  #5 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,731 Thanks: 707  
May 16th, 2017, 08:22 AM  #6 
Newbie Joined: May 2017 From: alabama Posts: 6 Thanks: 0 
So if the question read A tank of water is being drained through an outlet, the height H (m) of water in the time t (s) is given by dh/dt =  (0.003) √H given t = 0 and H = 4 determine an expression for H in terms of t. That would give the same answer? 

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