Chapter 2, Problem 3P

To determine

(a)

The volume of traffic across the bridge.

Answer to Problem 3P

The volume of traffic across the bridge is $333.33\text{veh}/\text{hr}$.

Explanation of Solution

Given:

The total cost to travel across bridge except tolls is expressed as,

  $C=50+0.5V$.

Here, $V$ is the number of vehicles per hour and $C$ is the $\text{cost}/\text{veh}$ in cents.

The demand to travel across the bridge is,

  $V=2500-10C$.

Calculations:

Calculate volume of traffic across the bridge.

The total cost to travel across bridge except tolls is,

  $C=50+0.5V$ ....... (I)

The volume across the bridge is,

  $V=2500-10C$ ....... (II)

Solve Equations (I) and (II).

  $C=216.67$

  $V=333.33$

Conclusion:

Therefore, the volume of traffic across the bridge is $333.33\text{veh}/\text{hr}$.

To determine

(b)

The volume of traffic across the bridge if a toll of $25\text{cents}$ is added and for $50\text{cents}$ increase.

Answer to Problem 3P

The volume of traffic across the bridge if a toll of $25\text{cents}$ is added is $291.67\text{veh}/\text{hr}$ and for $50\text{cents}$ increase is $327.49\text{veh}/\text{hr}$.

Explanation of Solution

Concept used:

Write the expression to calculate the volume expected for a $50\text{cents}$ increase.

  $V^{\prime }=V-\left(\frac{V_{\text{new}}}{50}\right)$

Here, $V_{\text{new}}$ is the volume of traffic across the bridge if toll is increased to $25\text{cents}$ and $V$ is the volume of traffic across the bridge.

Calculations:

Calculate volume of traffic across the bridge if toll is increased to $25\text{cents}$.

The new cost of the cost of travel across bridge is,

  $C=50+0.5V+25$

Substitute $\left(50+0.5V+25\right)$ for $C$ in Equation (II).

  $\begin{array}{c}V=2,500-10\left(50+0.5V+25\right)\\ V=2,500-500-5V-250\\ 6V=1,750\\ V=291.67\end{array}$

Calculate volume of traffic expected for a $50\text{cents}$ increase.

Substitute $333.33\text{veh}/\text{hr}$ for $V_{\text{new}}$ and $291.67\text{veh}/\text{hr}$ for $V$ in Equation (II).

  $\begin{array}{c}{V}^{\prime }=333.33\text{veh}/\text{hr}-\left(\frac{291.67\text{veh}/\text{hr}}{50}\right)\\ =333.33\text{veh}/\text{hr}-5.8334\text{veh}/\text{hr}\\ =327.49\text{veh}/\text{hr}\end{array}$

Conclusion:

Therefore, the volume of traffic across the bridge if a toll of $25\text{cents}$ is added is $291.67\text{veh}/\text{hr}$ and for $50\text{cents}$ increase is $327.49\text{veh}/\text{hr}$.

To determine

(c)

The volume of traffic across the bridge.

Answer to Problem 3P

The volume of traffic across the bridge is $666.67\text{veh}/\text{hr}$.

Explanation of Solution

Given:

The total cost to travel across bridge including toll is expressed as,

  $C=50+0.2V$

Calculations:

Calculate volume of traffic across the bridge.

Substitute $\left(50+0.2V\right)$ for $C$ in equation (II).

  $\begin{array}{c}V=2,500-10\left(50+0.2V\right)\\ V=2,500-500-2V\\ 3V=2,000\\ V=666.67\end{array}$

Conclusion:

Therefore, the volume of traffic across the bridge is $666.67\text{veh}/\text{hr}$.

To determine

(d)

The toll to yield the highest revenue for demand and supply function and the associated demand and revenue.

Answer to Problem 3P

The toll which yield the highest revenue for demand and supply function is $\$1.00$, associated demand is $166.67\text{veh}/\text{hr}$ and revenue. $\$16,666,67$ per hour.

Explanation of Solution

Concept used:

Write the expression to calculate the revenue generated.

  $R=V\times T$ ....... (III)

Here, $R$ is the revenue and $T$ is the toll.

Calculations:

Assume the toll rate as $T$ to yield highest revenue.

The new cost of the cost of travel across bridge is,

  $C=50+0.5V+T$

Substitute $\left(50+0.5V+T\right)$ for $C$ in Equation (II).

  $\begin{array}{c}V=2,500-10\left(50+0.5V+T\right)\\ V=2,500-500-5V-10T\\ 6V=2,000-10T\\ V=\frac{2,000-10T}{6}\end{array}$ ....... (IV)

Substitute $\left(\frac{2,000-10T}{6}\right)$ for $V$ in Equation (III).

  $\begin{array}{r}R=\left(\frac{2,000-10T}{6}\right)\times T\\ =\frac{2,000T-10T^2}{6}\end{array}$ ....... (V)

From the above equation, the toll which would yield the maximum revenue is 100 cents or $\$1.00$.

Substitute $100$ for $T$ in Equation (V).

  $\begin{array}{c}R=\frac{2,000\left(100\right)-10{\left(100\right)}^2}{6}\\ =\frac{2,00,000-1,00,000}{6}\\ =\frac{1,00,000}{6}\\ =16,666,67\end{array}$

Calculate the demand for travel across the bridge for maximum revenue.

Substitute $100$ for $T$ in equation (IV).

  $\begin{array}{c}V=\frac{2,000-10\left(100\right)}{6}\\ =\frac{2,000-1,000}{6}\\ =\frac{1,000}{6}\\ =166.67\end{array}$

Conclusion:

Therefore, the toll which yield the highest revenue for demand and supply function is $\$1.00$, associated demand is $166.67\text{veh}/\text{hr}$ and revenue. $\$16,666,67$ per hour.