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March 25th, 2017, 02:02 AM  #1 
Newbie Joined: Mar 2017 From: Scotland Posts: 8 Thanks: 0  Determine the Minimum value of P
Hello, Sorry if this is in the wrong section; I don't know the difference between calculus and precalculus I'm having difficulties with a question of mine: The cost Â£P million, of laying 1km of pipe for a water main is given by the formula P = 4000/9a + 4a where a is the crosssectional area of the pipe in square metres. Use calculus to find the minimum value of P, justifying your answer. I understand how to differentiate it and making the derivative equal to 0 to get the maximum and minimum points but I'm struggling to solve for a and any further point after that. Thanks in advance, Any help is greatly appreciated. 
March 25th, 2017, 05:53 AM  #2 
Senior Member Joined: Jun 2015 From: England Posts: 676 Thanks: 194 
$\displaystyle P = \frac{{4000}}{9}{a^{  1}} + 4a$ $\displaystyle \frac{{dP}}{{da}} =  \frac{{4000}}{9}{a^{  2}} + 4 = 0$ Is what sort of equation which has how many roots? So how do you use calculus to tell whether the root is a max or a min? 
March 26th, 2017, 02:28 AM  #3  
Newbie Joined: Mar 2017 From: Scotland Posts: 8 Thanks: 0  Quote:
a^2 = 9/1000 I've no idea how to solve for a then finding the minimum value.  
March 26th, 2017, 03:36 AM  #4 
Senior Member Joined: Jun 2015 From: England Posts: 676 Thanks: 194 
$\displaystyle 4  \frac{{4000}}{9}{a^{  2}} = 0$ multiply through by $\displaystyle {a^2}$ $\displaystyle 4{a^2}  \frac{{4000}}{9} = 0$ $\displaystyle {a^2} = \frac{{1000}}{9}$ This is a simple quadratic equation in a, can you not solve it for two values of a? What does the fact that dP/da = 0 mean? It means that the slope of the curve is zero. This happens at both a minimum and a maximum, but we don't know which. To find out we take the second derivative $\displaystyle If\quad \frac{{{d^2}P}}{{d{a^2}}}\quad is\;negative\;the\;po{\mathop{\rm int}} \;is\;a\;\max imum$ $\displaystyle If\quad \frac{{{d^2}P}}{{d{a^2}}}\quad is\;positive\;the\;po{\mathop{\rm int}} \;is\;a\;\min imum$ https://en.wikipedia.org/wiki/Derivative_test 
March 26th, 2017, 04:00 AM  #5 
Newbie Joined: Mar 2017 From: Scotland Posts: 8 Thanks: 0 
Alright, would this be correct? I've gotten a = (10âˆš10)/3 and a= (10âˆš10)/3 I've found the second derivative to be: 2(4000/9)a^3 When I sub in a I get 0.7589 which is greater than 0 which implies a maximum. Therefore P = Â£0.76 Million? Is that it? Also how do you use MathML? What I'm typing looks really ugly. 
March 26th, 2017, 04:22 AM  #6 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,348 Thanks: 461 Math Focus: Yet to find out.  
March 26th, 2017, 04:26 AM  #7 
Math Team Joined: Jul 2011 From: Texas Posts: 2,639 Thanks: 1319 
Note that $a > 0$ in the given context of this problem, so you can ignore the negative solution. 2nd derivative > 0 indicates a minimum ... 
March 26th, 2017, 04:34 AM  #8  
Newbie Joined: Mar 2017 From: Scotland Posts: 8 Thanks: 0  Ahh, I meant to write minimum, I have no idea what I was thinking when I typed that. Quote:
Thanks for the help with this problem everyone, I understand it much better.  

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