HW1a_Solutions

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Oregon State University, Corvallis *

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Civil Engineering

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Apr 3, 2024

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Homework 1- Solutions January 4, 2021 CE 392 Introduction to Highway Engineering Chapter 1 Garber and Hoel These 3 problems are open-ended Problems 1-11 1-14 1-15 Chapter 2 Garber and Hoel Problems 2-1 2-3 2-4 2-5 2-7 2-13 2-16

Homework 1- Solutions January 4, 2021 2-1 ( 5 points ) How would your typical day be changed without availability of your principal mode of transportation? Consider both personal transportation as well as goods and services that you rely on. Example answer: A typical day in my life would be significantly different without the airplane. Although I do not use this mode daily, goods and services that I do purchase are transported via this mode. Other modes, such as trucking, trains, and the automobile, could serve as replacements to the airplane; however, the airplane significantly lowers the transit time for shipping goods. For example, I mail a letter to California. Typical transit time for this letter using the airplane is three days. By using another mode other than the airplane, the transit time for the same letter would probably exceed seven days. As for my personal transportation, long distance travel is accomplished by using the airplane. For example, I take a vacation to Europe. If I travel using a cruise ship, it would take me in excess of seven days to reach Europe. However, if I fly, I can arrive in Europe within nine hours. Having the ability to transport people and goods quickly allows the international trade market to prosper, which in turn provides me with goods in a timely and efficient manner.

Homework 1- Solutions January 4, 2021 2-3 ( 20 points ) 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 + 0.5V, where V is the number of veh/hr and C is the cost/vehicle in cents. The demand for travel across the bridge is V = 2500 −10C . (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 tollbooth is to be added, thus reducing the travel time to cross the bridge. The new cost function is C = 50 + 0.2V. Determine the volume of traffic that 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. (a) Determine the volume of traffic across the bridge. Substitute the total cost function into the demand function and solve for V. V = 2500 –10(50 + 0.5V) V = 2500 – 500 – 5V 6V = 2000 V = 333.33 vehicles/ hour Therefore, the number of vehicles wanting to cross this bridge is 334 vehicles/hour. (b) If a 25 cent toll was added, what is the new volume crossing the bridge? 1. Add 25 cents to the original cost function. C = 50 + 0.5V + 25 C = 75 + 0.5V Substitute the above cost function into the demand function and solve for V. V = 2500 – 10(75 + 0.5V) V = 2500 – 750 – 5V 6V = 1750 V = 291.667 Therefore, the new volume crossing the bridge will now be 292 vehicles / hour with a 25 cents toll.

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Homework 1- Solutions January 4, 2021 (c) An additional toll booth changed the cost function to C = 50 + 0.2V. Determine the new volume of vehicles wanting to cross this bridge. Substitute the new cost function into the demand function and solve for V. V = 2500 –10 (50 + 0.2V) V = 2500 – 500 – 2V 3V = 2000 V = 666.67 vehicles/ hour Therefore, the new number of vehicles wanting to cross this bridge is 667 vehicle/hour. (d) Determine the toll to yield the highest revenue for part a. Assume 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 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 , substitute the above expression into the revenue formula and differentiate with respect to T. R = T * ((2000 - 10T) / 6) R = (2000T - 10T 2 ) / 6 dR/dT (2000T - 10T 2 ) / 6 = 0 (2000 - 20T) / 6 = 0 Therefore, the toll which would yield the maximum revenue is T = 100, or T = $1.00. R = T * V R = (2000T - 10T 2 ) / 6 R = (2000(100) - 10(100) 2 ) / 6 R = 16,666.67 Therefore, a toll of $1.00 will yield a revenue of $166.67 per hour.

Homework 1- Solutions January 4, 2021 2-4 ( 20 points ) A toll bridge carries 10,000 veh/day. The current toll is $3.00/vehicle. Studies have shown that for each increase in toll of 50 cents, the traffic volume will decrease by 1000 veh/day. It is desired to increase the toll to a point where revenue will be maximized. (a) Write the expression for travel demand on the bridge, related to toll increase and current volume. (b) Determine toll charge to maximize revenues. (c) Determine traffic in veh/day after toll increase. (d) Determine total revenue increase with new toll. (a) Write the expression for travel demand on the bridge. Let V = travel demand. V = 10000 - 1000( x / 50) (b) Determine the toll charge to maximize revenues. Let T = toll charge. Since the original toll was 300 cents per vehicle, the new toll charge will be T = 300 + 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 = (10000 - 1000( x / 50)) * (300 + x ) R = (10000 - 20 x ) * (300 + x ) R = 3000000 + 10000 x - 6000 x - 20 x 2 dR/dx (3000000 + 4000 x - 20 x 2 ) = 0 4000 - 40 x = 0 x = 100 Therefore, an increase in toll of 100 cents will maximize revenues. (c) Determine the traffic volume per day after the toll increase. Now, substitute the new toll, x, into the demand function developed in part a. V = 10000 - 1000( x / 50) V = 10000 - 1000(100/50) V = 10000 - 2000 V = 8000 vehicles per day The new demand for the bridge will be 8,000 vehicles per day. (d) Determine the total revenue with the new toll. R = V * T R = 8000 * (300 + 100) R = 8000 * 400 R = $3,200,000 The total revenue to be generated with the new toll will be $3,200,000 per day.

Homework 1- Solutions January 4, 2021 2-5 ( 20 points ) 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? 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 + 4500 x - 2000 x - 16 x 2 dR/dT (562,500 + 2500 x - 16 x 2 ) = 0 32 x = 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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Homework 1- Solutions January 4, 2021 2-7 ( 20 points ) An individual is planning to take an 800-mile trip between two large cities. Three possibilities exist: air, rail, or auto. The person is willing to pay $25 for every hour saved in making the trip. The trip by air costs $600 and travel time is 8 hrs, by rail the cost is $450 and travel time is 16 hrs, and by auto the cost is $200 and travel time is 20 hrs. (a) Which mode is the best choice? (b) What factors other than cost might influence the decision regarding which mode to use? Determine the total cost (initial cost plus time cost) for each mode. Total Cost (air) = 600 + (8 * 25) Total Cost (air) = $800 Total Cost (rail) = 450 + (16 * 25) Total Cost (rail) = $850 Total Cost (auto) = 200 + (20 * 25) Total Cost (auto) = $700 From the above analysis, it appears that the best mode to choose to make this trip is automobile . Other factors to consider, other than costs, when selecting a mode to travel might include the following: personal comfort the modes have to offer, whether additional connections need to be made to reach the final destination, the level of stress that can be anticipated by traveling by that mode, or whether the reason for travel is for business or pleasure.

Homework 1- Solutions January 4, 2021 2-13 ( 5 points ) Consult with the U.S. Department of Transportation website and identify the name and location of highways in your state that are included as part of the National Highway System. NOTE: Make sure the that the interstate or highways meet the criteria of being the National Highway System. 2-16 ( 10 points ) What do the following acronyms mean? AAA, AAR, AASHTO, AI, APTA, ARTBA, FHWA, PCA, TRB Acronym Definition AAA American Automobile Association AAR Association of American Railroads AASHTO American Association of State Highway and Transportation Officials AI Asphalt Institute APTA American Public Transit Association ARTBA American Road and Transportation Builders Association FHWA Federal Highway Administration PCA Portland Cement Association TRB Transportation Research Board

HW1a_Solutions

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