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Homework Help: Applications of Partial Derivatives

  1. Oct 8, 2015 #1
    1. The problem statement, all variables and given/known data
    Let l, w, and h be the length, width and height of a rectangular box. The length l is increasing with time at at rate of 1 m/s, while the width and the height are decreasing at rates 2 m/s and 1m/s respectively. At a certain moment in time the dimensions of the box are l=5, w=4m and h=3m. Find the rate of change of the volume of the box at this moment in time. Help please?

    2. Relevant equations
    The rate of change is just the derivative, but I am not sure how to write it out with 3 variables, I'm kind of stuck from the beginning.

    Chain rule

    3. The attempt at a solution

    I drew the box and labeled it l=5, w=4, h=3. I'm not sure how to proceed . . .
  2. jcsd
  3. Oct 8, 2015 #2


    Staff: Mentor

    The dimensions should be l, w, and h. Each dimension is changing in time; i.e., is a function (single-variable) of time. The values you show are the dimensions at a particular moment in time.
    What you need are the following:
    • A formula for the volume of the box at any time, not just when l = 5, w = 4, and h = 3.
    • The total derivative.
    Since you titled this thread "Applications of Partial Derivatives" there should be an example or two that shows how to apply the total derivative (which entails the use of partial derivatives).
  4. Oct 8, 2015 #3
    If the volume of the box is V(t) at time t, and the length, width, and height of the box are l(t), w(t), and h(t), how is V related to l, w, and h? Do you know how to use the product rule to find the derivative of V with respect to t as a function of l(t), w(t), and h(t) and their time derivatives?

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