Optimizing Tunnel Throughput: What Speed Maximizes Vehicle Volume Per Hour?

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Homework Statement


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?

Homework Equations


x = .324/(42.1-v)

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.
 

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Perplexed553 said:

Homework Statement


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?

Homework Equations


x = .324/(42.1-v)

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.

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

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.

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.
 
Perplexed553 said:
They show a solution where Q= cars/hour = (42.1v-v^2) / .324. I don't see where they get this.
Read vehicle flow analysis
that's where the formula comes from
 
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SabemosqueSePuede said:
Read vehicle flow analysis
that's where the formula comes from
In the homework exercise in post #1, the student was expected to derive that formula from the given one, x = 0.324/(42.1 - v).
Anyway, the thread is over six years old.