Little Math Question
<div class="IPBDescription">Exam tommorow :O</div> Yes I know there are probably better places to ask this (in fact I did try a Yahoo maths chatroom, only to have my question responded to by "A/S/L" "You male or female?" "A/S/L"), but I know you are all a (mostly) intellegent group of people, so you should find this one a doddle.
It's more of a general question than a specific one, I need to know how the differential of an equation relates to it's minimum value. For example,
A ship is to make a voyage of 100km at a constant speed of vkm/h.
The running cost of the ship is £(0.8v^2 + 2000/v) per hour.
(I) Write down the time taken to go 100km at vkm/h (100/v, easy)
(II) Hence write down the total cost of travelling 100km at vkm/h (80v + 200,000/v^2)
(III) By considering dC/dV, find the speed which keeps the cost of the journey to a minimum (to nearest km/h)
So... how do I do it? If I did it right, dC/dv is 80-400,000^-3 (I took 200,000/v^2 to be 200,000v^-2, is that right?)
But the value of 80-400,000^-3 is just 80, which is NOT the answer.
Please help a maths nub.
*edit* I'm starting to realise the error lies in the fact that I took 200,000/v^2 to be the same as 200,000v^-2. I thought the rules were 1/x = x^-1, and 2/x = 2x^-1 ? Am i wrong? *edit*
(Feel free to delete this once I get a correct answer, and I apologise for using the word "doddle")
It's more of a general question than a specific one, I need to know how the differential of an equation relates to it's minimum value. For example,
A ship is to make a voyage of 100km at a constant speed of vkm/h.
The running cost of the ship is £(0.8v^2 + 2000/v) per hour.
(I) Write down the time taken to go 100km at vkm/h (100/v, easy)
(II) Hence write down the total cost of travelling 100km at vkm/h (80v + 200,000/v^2)
(III) By considering dC/dV, find the speed which keeps the cost of the journey to a minimum (to nearest km/h)
So... how do I do it? If I did it right, dC/dv is 80-400,000^-3 (I took 200,000/v^2 to be 200,000v^-2, is that right?)
But the value of 80-400,000^-3 is just 80, which is NOT the answer.
Please help a maths nub.
*edit* I'm starting to realise the error lies in the fact that I took 200,000/v^2 to be the same as 200,000v^-2. I thought the rules were 1/x = x^-1, and 2/x = 2x^-1 ? Am i wrong? *edit*
(Feel free to delete this once I get a correct answer, and I apologise for using the word "doddle")
Comments
EDIT: Beh, I was differentiating the initial cost equation, and looked at your answer further down, assuming it was the same equation - I thought you had differentiated the term 0.8v^2 as 80.
<!--emo&;)--><img src='http://www.unknownworlds.com/forums/html/emoticons/wink.gif' border='0' style='vertical-align:middle' alt='wink.gif'><!--endemo-->
Um... huh? You differentiate 80v and you get 80 surely? Could you explain <i>which</i> 80 and why it should be 1.6v?
*edit* The "^" symbol means "to the power of"
No it isn't. You try it.
2/10 = 0.2
10^-2 = 0.01
2/X is 2x^-1 afaik.
*edit* Thank you madcow. Please don't confuse me on the night before my exam boggle <!--emo&:)--><img src='http://www.unknownworlds.com/forums/html/emoticons/smile.gif' border='0' style='vertical-align:middle' alt='smile.gif'><!--endemo--> *edit*
thanks for clearing that one
(how could i be so stupid...)
I believe when you typed this out you put down the wrong answer for #2. From your work on #3 it seems like you got the right idea though.
I believe that should be 80v + 200,000/(v^2) or 80v + 200,000v^-2.
Ok for number 3.
C = 80v + 200,000v^-2
Differentiate that and we get:
dC/dv = 80 - 400,000v^-3
I'm thinking that the lowest cost is the zero of this function, which is 17.099759 or about 17.
I'm pretty sure that this is right now. I think you differentiated 400,000v^-2 and got 400,000^-3. The v needs to stay in that equtation <!--emo&::nerdy::--><img src='http://www.unknownworlds.com/forums/html/emoticons/nerd.gif' border='0' style='vertical-align:middle' alt='nerd.gif'><!--endemo-->
I'm pretty sure that this is right now. I think you differentiated 400,000v^-2 and got 400,000^-3. The v needs to stay in that equtation <!--emo&::nerdy::--><img src='http://www.unknownworlds.com/forums/html/emoticons/nerd.gif' border='0' style='vertical-align:middle' alt='nerd.gif'><!--endemo--> <!--QuoteEnd--> </td></tr></table><span class='postcolor'> <!--QuoteEEnd-->
Don't you need to identify the turning points and stuff?
pei
I'm pretty sure that this is right now. I think you differentiated 400,000v^-2 and got 400,000^-3. The v needs to stay in that equtation <!--emo&::nerdy::--><img src='http://www.unknownworlds.com/forums/html/emoticons/nerd.gif' border='0' style='vertical-align:middle' alt='nerd.gif'><!--endemo--> <!--QuoteEnd--></td></tr></table><span class='postcolor'><!--QuoteEEnd-->
Don't you need to identify the turning points and stuff? <!--QuoteEnd--> </td></tr></table><span class='postcolor'> <!--QuoteEEnd-->
The zero of the differentiated function IS the turning point of the original function.
I'm pretty sure that this is right now. I think you differentiated 400,000v^-2 and got 400,000^-3. The v needs to stay in that equtation <!--emo&::nerdy::--><img src='http://www.natural-selection.org/forums/html/emoticons/nerd.gif' border='0' style='vertical-align:middle' alt='nerd.gif'><!--endemo--> <!--QuoteEnd--></td></tr></table><span class='postcolor'><!--QuoteEEnd-->
Don't you need to identify the turning points and stuff? <!--QuoteEnd--></td></tr></table><span class='postcolor'><!--QuoteEEnd-->
The zero of the differentiated function IS the turning point of the original function. <!--QuoteEnd--></td></tr></table><span class='postcolor'><!--QuoteEEnd-->
He's right you know. Mad Cow, you are right in every respect.
Yes, the answer to (II) should have been 80v + 200,000/v^2
Yes, I did forget the v. I wrote it pretty small and didn't notice it next to all those zeros....
And finally, yes, the answer is 17.
Want to have a go at the last part?
(IV) Find the minimum cost of the voyage.
*edit* Arg, ok, so I have 400,000v^-3 = 80, then I can divide to get v^-3 = 0.0625, but if I take the -3rd root, I don't get 17, I get -0.3968. What am I doing wrong? *edit*
80/400,000 = .0002
.0002^-1 = 5000
5000^(1/3) = 17.0997
That's the way I worked it at least.
Surely that would just give v^-4 = 2^-5?
And where did the ^(1/3) come from?
I know you're right, I just can't see the logic behind it.
400,000v^-3 = 80
v^-3 = 80/400,000
v^-3 = .0002
(v^-3)^-1 = .0002^-1
v^3 = 5000
(v^3)^(1/3) = 5000^(1/3)
v = 17.0997
I raised everything to the -1 to get rid of the negative power, when working problems like this I always try to get rid of negatives because they usually end up messing me up somewhere. This is just the way I think and therefore the way that I solved the problem. Raising a number to 1/3 power is like finding the 3rd root, just like raising a number to the 1/2 power is like finding the square root.
<!--QuoteBegin--></span><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin--> I probably won't have time to read your reply tonight, but why did you do 0.0002^-1?
Surely that would just give v^-4 = 2^-5?<!--QuoteEnd--></td></tr></table><span class='postcolor'><!--QuoteEEnd-->
From the above you can see that I did (v^-3)^-1. To simplify this you multiply the exponents so we have -3 * -1 = 3.
Doesnt finding the minimum invole the second differential... or am i thinking about whether its a min or max???
this actually got me thinking, but i dont know how close this is to the maths exam im doing... What country you in (if UK what yr/lvl/module you doing)... spose i better get some sleep... i got 2 maths exams tomorrow..... :/
So for the last part of your prob, you found the speed to mimimize cost (17.1 km/hr), and you just plug it back into your equation for total cost ie:
C(v) = 80*v + 200,000*v^-2 = 80*(17.1) + 200,000*(17.1)^-2 = 2051 (i think, i dont have a decent calculator around right now)
<!--QuoteBegin--></span><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin--><i>Mathematics is the supreme intellectual achievement and the most original creation of the human spirit</i> - Morris Kline.<!--QuoteEnd--></td></tr></table><span class='postcolor'><!--QuoteEEnd-->
<!--QuoteBegin--></span><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin--><i>Mathematics is the Queen and servant of the Sciences</i> - Karl Friedrich Gauss<!--QuoteEnd--></td></tr></table><span class='postcolor'><!--QuoteEEnd-->
<!--QuoteBegin--></span><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin--><i>Mathematics is just more and more complicated ways of saying 'Pigs is pigs'</i> - Bertrand Russell<!--QuoteEnd--></td></tr></table><span class='postcolor'><!--QuoteEEnd-->
To each his own I guess <!--emo&:D--><img src='http://www.unknownworlds.com/forums/html/emoticons/biggrin.gif' border='0' style='vertical-align:middle' alt='biggrin.gif'><!--endemo-->
Wrong the answer is 42.
I'm still waiting for one of my math problems to equal 42 or 133.7 or 13.37 or 1.337... you get the picture. I will be a very happy man when I notice I have a l33t answer.
Oh and yes I should just post my math homework here some time. It would get things done much faster. Better yet we need a homework forum. It would be fun for all.