Determine the toll charge to maximize revenues. Let T = toll charge.
A toll bridge carries 6,000 veh/day. The current toll is $3.50/vehicle. Studies have shown that for each increase in toll of 50 cents, the traffic volume will decrease by 500 veh/day. It is desired to increase the toll to a point where revenue will be maximized.
What will be an ideal response?
Since the original toll was 350 cents per vehicle, the new toll charge will be
T = 350 + x
The revenue (R) is generated by the equation R = V × T. Substitute the above expressions into the revenue function and differentiate with respect to x, setting the derivative equal to zero.
R = (6000 – 500(x / 50)) × (350 + x)
R = (6000 – 10x) × (350 + x)
R = 2100000 + 6000x – 3500x – 10x2
dR/dx (2100000 + 2500x – 10x2) = 0
2500 – 20x = 0
x = 125
Therefore, an increase in toll of 125 cents will maximize revenues.
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