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

[Perplexed553]

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.

 

[SteamKing]

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.

 

[haruspex]

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

 

[SabemosqueSePuede]

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

 

[haruspex]

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.

 

[Mark44]

Anyway, the thread is over six years old.

And deserves to be closed. The OP posted the question and never returned.