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Homework Help: Maximixe tunnel throughput

  1. Oct 5, 2014 #1
    1. The problem statement, all variables and given/known data
    historical data shows that between 15 and 35 mph, the space x between vehicles (in miles) is x = 0.324/(42.1 - v)
    where v is the vehicles speed in miles per hour

    Ignoring the length of individual vehicles, what speed will give the tunnel the largest volume in vehicles per hour?
    2. Relevant equations
    x = .324/(42.1-v)

    3. The attempt at a solution
    I thought I could set x = 0 (because spacing would be 0 ignoring the length of vehicles) and solve for v. That isn't correct.
    They show a solution where Q= cars/hour = (42.1v-v^2) / .324. I don't see where they get this.
    Then it looks like they take the derivative with respect to v of the numerator only and set it to next to 0
    dQ/dv = 0 = 42.1 -2v/.324 = 21.05 mph. This part makes sense to me but I still don't understand where they get Q = (42.1v - v^2)/.324.
  2. jcsd
  3. Oct 5, 2014 #2


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    If you analyze the formula for the spacing of the vehicles, you will see there is no speed v which can make x = 0.

    From the data given about vehicle speeds and vehicle spacing, you've got to figure out how to determine the number of vehicles Q which enter the tunnel, as a function of vehicle speed, v. Once you have Q as a function of vehicle speed, then you can figure out which speed v gives the maximum Q.
  4. Oct 6, 2014 #3


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    In case SteamKing's hint is not clear, there is a simple relationship between Q, v and x.
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