Results 1 to 5 of 5

Math Help - Partial Diff

  1. #1
    Junior Member
    Joined
    Feb 2008
    Posts
    61

    Partial Diff

    A closed rectangular box with unequal sides a, b, c has its edges slightly altered in length by amounts (delta a), (delta b), (delta c), respectively so that both its volume and surface area remained unchanged. Show that:

    (delta a)/a^2(c-b) = (delta b)/b^2(a-c) = (delta c)/c^2(b-a)


    How can i solve this question?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Dec 2007
    From
    Anchorage, AK
    Posts
    276
    This is about relating differentials. Our volume is V=abc and our surface area S=2(ab+bc+ac).
    Our volume and surface area do not change; this means dV=0 and dS=0.
    Now, dV=bc\,da+ac\,db+ab\,dc
    Thus bc\,da+ac\,db+ab\,dc=0
    Dividing by the volume abc, we get
    \frac{da}{a}+\frac{db}{b}+\frac{dc}{c}=0 (1)
    Also:
    dS=2(a\,db+b\,da+b\,dc+c\,db+a\,dc+c\,da)
    dS=2((b+c)da+(a+c)db+(a+b)dc)
    Thus:
    2((b+c)da+(a+c)db+(a+b)dc)=0
    (b+c)da+(a+c)db+(a+b)dc=0 (2)

    Solving (1) and (2) both for dc and combining:
    -\frac{(b+c)da+(a+c)db}{a+b}=dc=-c\left(\frac{da}{a}+\frac{db}{b}\right)
    ab((b+c)da+(a+c)db)=c(b(a+b)da+a(a+b)db)
    b^2(a-c)da=a^2(c-b)db
    Dividing by a^2b^2(c-b)(a-c), we get
    \frac{da}{a^2(c-b)}=\frac{db}{b^2(a-c)}

    Similar solving of (1) and (2) for da and db gives the other equalities.

    -Kevin C.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Feb 2008
    Posts
    61
    why do i have to divide by V=abc to get eqn (1)?
    is it a type of differentiation rule?
    or

    i suppose its a way of simplification?===> dividing both side by abc
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by MilK View Post
    why do i have to divide by V=abc to get eqn (1)?
    is it a type of differentiation rule?
    or

    i suppose its a way of simplification?===> dividing both side by abc Mr F says: Yes.
    Beaten to the punch. I'd written it on a napkin and was just about to type it in. Probably all for the best since I spilt soup du jour on the napkin - some soup spots can look just like differentials .....
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Feb 2008
    Posts
    61


    Thanks guys
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Partial diff
    Posted in the Calculus Forum
    Replies: 5
    Last Post: June 29th 2011, 04:18 PM
  2. Partial Diff.
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 17th 2008, 04:23 PM
  3. Partial Diff(3)
    Posted in the Calculus Forum
    Replies: 18
    Last Post: February 24th 2008, 03:20 AM
  4. Partial Diff (2)
    Posted in the Calculus Forum
    Replies: 6
    Last Post: February 16th 2008, 05:10 PM
  5. partial diff.
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 24th 2007, 12:05 PM

Search Tags


/mathhelpforum @mathhelpforum