Expert: Paul Klarreich - 9/21/2008

Question
Paul, I am seeking to use the following Daniels/Gilbert formula which uses velocity (m/min) and time (min to complete a race) to calculate a pseudo VO2 Max:  VO2 Max=(-4.60 + 0.182258 * velocity + 0.000104 * velocity^2)/(0.8 + 0.1894393 * e^(-0.012778 * time) + 0.2989558 * e^(-0.1932605 * time)) I've sources it from: http://www.runningforfitness.org/faq/vo2.php   I would like to rearrange the formula so that with a given VO2Max, I can predict a runner’s finish time.    Thanks, Steve

Answer
Questioner:   Steve
Category:  Advanced Math
Private:  No
 
Subject:  Rearrange a Formula
Question:  Paul, I am seeking to use the following Daniels/Gilbert formula which uses velocity (m/min) and time (min to complete a race) to calculate a pseudo VO2 Max:  VO2 Max=(-4.60 + 0.182258 * velocity + 0.000104 * velocity^2)/(0.8 + 0.1894393 * e^(-0.012778 * time) + 0.2989558 * e^(-0.1932605 * time)) I've sources it from: http://www.runningforfitness.org/faq/vo2.php   I would like to rearrange the formula so that with a given VO2Max, I can predict a runner’s finish time.    Thanks, Steve
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Hi, Steve,

I generally reject questions that require specialized jargon, but your question basically says:

Solve for t:

                    -4.60 + 0.182258v + 0.000104v^2
M =     -----------------------------------------------------------------
        0.8 + 0.1894393 * e^(-0.012778t) + 0.2989558 * e^(-0.1932605t)

I have approximated some of your numbers to give:

            -4.60 + 0.18 v + 0.0001 v^2
M =     ------------------------------------
        0.8 + 0.2 e^-0.013t + 0.38 e^-0.2t


Alas, there isn't any easy way I see to solve this.  You want:

     
                                         -4.60 + 0.18 v + 0.0001 v^2
0.8 + 0.2 e^-0.013t + 0.38 e^-0.2t =     -----------------------------
                                                        M

For a given M and v, you can compute the right side.  After that, you will need some numerical (computational) method to approximate a solution.  [Look up Newton's method.]