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Application of derivatives problem?

  1. Mar 5, 2014 #1
    1. The problem statement, all variables and given/known data

    A biologist determines experimentally that the number of calories burned by a salmon swimming a distance d in miles upstream against a current v0 in miles per hour is given by

    Energy = kdv^5/v − v0

    where v is the salmon’s swimming speed relative to the water it is in. This means that the salmon’s progress upstream is at the rate of v − v0 miles per hour, so that the distance d is covered in a time of

    t=d/v − v0

    If v0 = 2 mph and d = 20 miles, and the salmon, being smart, swims so as to minimize the calories burned, how many hours will it take to complete the journey?

    2. Relevant equations

    Requires derivatives

    3. The attempt at a solution

    I really just don't understand where to start. I subbed t into the equation and ended up with ktv^5/v-v0, took the derivative of that with respect to t and got 5kv^4 but I have no idea what I really don't know where to go from there.
  2. jcsd
  3. Mar 5, 2014 #2


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    Do you mean ##\frac{kdv^5} v - v_0##, which is what you wrote, or ##\frac{kdv^5}{v − v0}##?

    Same problem here. Obviously you mean ##\frac d {v-v_0}##. Use parentheses!!
    Try finding the critial value of ##v## using the energy equation. Then use that to get the time of travel.
  4. Mar 5, 2014 #3
    Yes! I meant (kdv^5)/(v−v0).
  5. Mar 5, 2014 #4
    OK, so I'm guessing I don't plug t into the energy equation in the first place? because I did that and found the derivative to be 5ktv^4
  6. Mar 5, 2014 #5


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    Why are you guessing? Why not just try what I suggested?
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