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Implicit function

  1. Oct 3, 2015 #1
    1. The problem statement, all variables and given/known data
    The length ℓ, width w, and height h of a box change with time. At a certain instant the dimensions are ℓ = 4 m and
    w = h = 9 m, and ℓ and w are increasing at a rate of 1 m/s while his decreasing at a rate of 6 m/s. At that instant find the rates at which the following quantities are changing.

    (A) The Volume
    (B) The Surface Area
    (C) the length of a diagonal (round two decimals places)

    2. Relevant equations
    The Chain rule, Partial Derivative

    3. The attempt at a solution
    I already found A (ans: -99 m^3/s) and B (ans: -94 m^2/s)
    for C:
    1. i differentiated the formula L^2= ℓ^2+w^2+h^2 to 2L(dL/dt) = 2ℓ(dℓ/dt) + 2w(dw/dt) + 2h (dh/dt)
    2. let dℓ/dt = dw/dt = 1 m/s and dh/dt = -6 m/s
    3. my answer came as 2L(dL/dt) = 82 but i'm completely lost after this part
  2. jcsd
  3. Oct 3, 2015 #2


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    At time t, you know what the values of w, h, and l are. You want to solve 2L (dL/dt) = 82 for dL/dt.
  4. Oct 3, 2015 #3
    would i be able to find L by square rooting (w,h,ℓ) and then multiplying it with the 2?

    update: i manage to figure out the equation by doing just that. Thanks for your help!
    Last edited: Oct 3, 2015
  5. Oct 4, 2015 #4


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    You could but if you implicit differentiation you shouldn't! The length of the diagonal is given by [itex]L= (ℓ^2+ w^2+ h^2)^{1/2}[/itex].
    Last edited by a moderator: Oct 4, 2015
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