Chapter 2, Problem 4P

To determine

(a)

The expression for travel demand on the bridge related to toll increase and current volume.

Answer to Problem 4P

The expression for travel demand on the bridge related to toll increase and current volume is $V=6000-\frac{x}{50}\left(500\right)$.

Explanation of Solution

Given:

A toll bridge carries $6000\text{veh}/\text{day}$.

The current toll is $\$3.50/\text{vehicle}$.

For each increase in toll of $50$ cents, the traffic volume will decrease by $500\text{veh}/\text{day}$.

Concept used:

Write the relation between traffic volume and cost.

  $V=V^{\prime }-\left(\frac{x}{x^{\prime }}\right)\Delta V$ ....... (I)

Here, $V^{\prime }$ is the vehicles carried by the bridge, $x$ is the cost increase in cents, $x^{\prime }$ is the increase in cost of toll and $\Delta V$ is the increase or decrease in the traffic volume.

Calculations:

Substitute $6000$ for $V^{\prime }$, $50$ for $x^{\prime }$ and $500$ for $\Delta V$ in equation (I).

  $V=6000-\frac{x}{50}\left(500\right)$

Conclusion:

Therefore, the expression for travel demand on the bridge related to toll increase and current volume is $V=6000-\frac{x}{50}\left(500\right)$.

To determine

(b)

The toll charge to maximize revenue.

Answer to Problem 4P

The toll charge to maximize the revenue is $\$4.75$.

Explanation of Solution

Write the expression to calculate the total toll charge.

  $T=\text{Original}\text{cost}+\text{Cost}\text{increase}$ ....... (II)

Here, $T$ is the toll.

Write the expression to calculate the revenue generated.

  $R=V\times T$ ....... (III)

Here, $R$ is the revenue.

Calculations:

Calculate the total toll charge.

Substitute $350$ for Original cost and $x$ for Cost increase in equation (II).

  $T=350+x$ ....... (IV)

Calculate volume of traffic expected for a 50 cent increase.

Substitute $\left(350+x\right)$ for $T$ and $\left[6000-\frac{x}{50}\left(500\right)\right]$ for $V$ in equation (III).

  $\begin{array}{c}R=\left(6000-\frac{x}{50}\left(500\right)\right)\times \left(350+x\right)\\ =\left(6000-10x\right)\left(350+x\right)\\ =2,100,000+6,000x-3,500x-10x^2\\ =2,100,000+2,500x-10x^2\end{array}$

For maximum value, calculate the first derivative of above equation and equate to zero.

  $\begin{array}{l}\frac{d}{dx}\left(2,100,000+2,500x-10x^2\right)=0\\ 2500-20x=0\\ 20x=2500\\ x=125\end{array}$

Substitute $125$ for $x$ in equation (IV).

  $\begin{array}{c}T=350+125\\ =475\text{cents}\left(\frac{1\$}{100\text{cent}s}\right)\\ =\$4.75\end{array}$

Conclusion:

Therefore, the toll charge to maximize revenue is $\$4.75$.

To determine

(c)

The traffic volume after toll increase.

Answer to Problem 4P

The traffic volume after toll increase is $4750\text{veh}/\text{day}$.

Explanation of Solution

Calculations:

Calculate the traffic volume after toll increase.

Consider the expression for travel demand on the bridge related to toll increase and current volume.

  $V=6000-\frac{x}{50}\left(500\right)$.

Substitute $125$ for $x$ in above equation.

  $\begin{array}{c}V=6,000-\frac{125}{50}\left(500\right)\\ =6,000-2.5\left(500\right)\\ =6,000-1,250\\ =4750\end{array}$

Conclusion:

Therefore, the traffic volume after toll increase is $4750\text{veh}/\text{day}$.

To determine

(d)

The total revenue increase with new toll.

Answer to Problem 4P

The total revenue increase with new toll is $\$22,562.5$.

Explanation of Solution

Calculations:

Substitute $4750\text{veh}/\text{day}$ for $V$ and $\$4.75$ for $T$ in equation (III).

  $\begin{array}{c}R=4750\times \$4.75\\ =\$22,562.5\end{array}$

Conclusion:

Therefore, the total revenue increase with new toll is $\$22,562.5$.