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Inverse Exponential Function

  1. Oct 2, 2012 #1
    1. The problem statement, all variables and given/known data

    When a camera flashes, the batteries begin recharging the flash capacitor which stores the charge Q according to the function Q(t) = Q* (1-e-t/a) where t is the elapsed time in seconds since the camera flash and Q* and a are non-zero

    (a) What does Q* represent?
    (b) Find the inverse of this function
    (c) How long does it take to recharge the capacitor to 90% if a = 5?

    2. Relevant equations



    3. The attempt at a solution

    I said that Q* was the constant of proportionality, but I'm not sure how right that is.

    My attempt at the inverse came out as t*a/ln(Q*) = Q

    I'm wondering if my answer for (a) is at all correct, and where to go from here, I have no idea how to attempt (c) and I think my answer to (b) is wrong.
     
  2. jcsd
  3. Oct 2, 2012 #2

    Ray Vickson

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    Try plotting Q(t) for a couple of different Q* values (but for the same a---just pick some value). What do you get?

    RGV
     
  4. Oct 2, 2012 #3
    We're not allowed to use a graphing calculator in this course, otherwise I would've played around with the graphs a bit, but I don't want to become dependent on my calculator to solve these, cause it'll bit me in the arse in the exam
     
  5. Oct 2, 2012 #4
    a. I believe that Q* should be some type of initial constant, and a is proportionality constant. I'm not entirely sure of this, but that's what I'm thinking just off of the top of my head. If it helps you, I'm thinking of this as a differential equation; I can explain more to you if you would like.

    b. The inverse should be:
    [itex]Q = Q(t)[/itex]

    [itex]t = Q[/itex]*[itex](1-e^{\frac{-Q}{a}})[/itex]

    [itex]1-\frac{t}{Q*}=e^{\frac{-Q}{a}}[/itex]

    [itex]ln(1-\frac{t}{Q*})=-Q*a[/itex]

    [itex]Q=\frac{-ln(1-\frac{t}{Q})}{a}[/itex]

    c. You plug in .9 for the value of Q(t), and 5 as the value for a. The value for Q* is 1 I believe, but I am not absolutely certain on that one. With these values, solve the equation for t.
     
  6. Oct 2, 2012 #5

    Ray Vickson

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    You are certainly allowed to draw graphs on pieces of scrap paper and throw them away later, after you have used them to gain insight into some issues. The point is that you should not need to draw a graph at all, but since you are missing a key insight, drawing a graph would be helpful to you---more helpful in the long run than being told by someone what is the correct answer.

    RGV
     
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