Consideration is being given to increasing the toll on a bridge now carrying 4500 veh/day. The current toll is $1.25/veh. It has been found from past experience that the daily traffic volume will decrease by 400 veh/day for each 25¢ increase in toll. Therefore, if x is the increase in toll in cents/veh, the volume equation for veh/day is V = 4500 - 400 (x / 25) , and the new toll/veh would be T = 125 + x. In order to maximize revenues, what would the new toll charge be per vehicle and what would the traffic in veh/day be after the toll increase?
What will be an ideal response?
First, solve for the revenue to be generated by the new toll.
R = V × T
R = (4500 – 400(x / 25)) × (125 + x)
R = (562,500 + 4500x – 2000x – 16x2
dR/dT (562,500 + 2500x – 16x2) = 0
32x = 2500
x = 78.125
Therefore, a toll increase of 78.125 cents per vehicle will maximize revenues for the bridge. For practical purposes and traveler convenience, round the toll increase to 75 cents.
Next, determine the resulting volume after the new toll increase. Simply substitute the new toll into the demand function above.
V = 4500 – 400(75/25)
V = 4500 – 1200
V = 3,300 vehicles per day
An increase in toll of 75 cents per vehicle will result in a new demand for the bridge of 3,300 vehicles per day.
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