A bridge has been constructed between the mainland and an island . the total cos

Question

A bridge has been constructed between the mainland and an island . the total cost ( excluding tolls ) to travel across the bridge is expressed as ( C = 50 + 0.5 V ) , where V : is the number of Veh/hr , and C : is the Cost/Veh in cents . the demand for travel across the bridge is ( V = 2500 - 10 C )

a) Determine the volume of traffic across the bridge .

b) If a toll of 25 cents is added , what is the volume across the bridge ?

c) A toolbooth is to be added , thus reducing the travel time to cross the bridge , the new cost function is ( C = 50 + 0.2 V ) , Determine the volume of traffic the would cross the bridge .

d) Determine the toll to yield the highest revenue for demand and supply function in part (a) and the associated demand and revenue .

Explanation / Answer

By substituting the value of cost (C) from equation (1) to equation (2), we get the following:

V=2500 -10(50+0.5V)

V=2500-500-5V

6V=2000

V=333.33 vehicles/hour » 334 vehicles/hour

Therefore, the number of vehicles wanting to cross this bridge is 334 vehicles/hour.

2.If a 25 cent toll was added, what is the new volume crossing the bridge?

Adding 25 cents to the original cost we get,

         C=50+0.5V+25

         C=75+0.5V

Now, we have to substitute the value of C into the demand function in order to get the value of V.

V=2500-10(75+0.5V)

V=2500-750-5V

6V=1750

V=291.667 » 292 vehicles/hour.

Therefore, the new volume crossing the bridge will now be 292 vehicles/hour.

   3.    An additional toll booth changed the cost function to C=50+0.2V. Determine the new volume of vehicles wanting to cross this bridge.

Substituting the value new Cost into the demand function we can get the value of V.

         V=2500-10(50+0.2V)

         V=2500-500-2V

         3V=2000

         V=666.67 » 667 vehicles/hour

Therefore, the new number of vehicles wanting to cross this bridge is 667 vehicles/hour.

                     4.To determine the toll to yield the highest revenue for part (a)

We assume a toll rate at T. The new cost function will be C=50+0.5V+T. Since the revenue generated is the toll rate, T, time the volume, V, first we have to solve for V with the new cost function.

            V=2500-10(50+0.5V+T)

            V=2500-500-5V-10T

            V=(2000-10T)/6

Since the revenue generated is R=T*V, we substitute the above expression into the revenue formula and differentiate with respect to T.

            R=T*((2000-10T)/6)

            R=(2000T-10T2)/6

Now,

            (2000-10*2T)/6=0

            T=100cents or $1.00

Therefore, T=$1.00 will yield the maximum revenue.

Now,   R=(2000T-10T2)/6

            R=(2000(100)-10(100)2)/6

            R=16,666.67 cents/hour » $166.67per hour

Therefore, a toll of $1.00 will yield a revenue of $166.67 per hour.

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