A contractor has a fleet of dump trucks for moving construction debris. The cost

Question

A contractor has a fleet of dump trucks for moving construction debris. The cost of running this fleet is dollar 325/hr per truck. For a given contract, a total of 2, 500 to:, must be removed from one construction site and moved to another location 35 miles away. The average round trip truck speed diminishes with its load by the equation, S = 40- 1.75L, where L is the truckload (in tons) and S is the truck speed (in miles per hour). Time is also required for loading the trucks. The time required to load a truck is given by F = 0.3L^0.5, where F is in hours. Develop a cost function for the total cost of removing the debris as a function of each truck's load. Using calculus, find the individual truckload, which minimizes the contractor's total cost for removing the debris. What is this total cost? (Be sure your solution is a minimum.) Does the distance the debris is moved affect this optimal loading? Does it affect the total cost? Justify your answer. How is this optimal loading changed if the maximum truck speed is changed from 35 to 45 mph or to 25 mph? What would these changes do to the cost of this particular job?

Explanation / Answer

Truck Load Speed Distance travelled in one round trip Time required for each round trip SqrtL Time for loading the Truck Total Time Required for 1 trip Total Load to be transfered Total Number of trips required Total Time required to transfer the complete debris Cost of running the fleet Total Cost for removing the debris L S=40-1.75*L D T=D/S F= 0.3*SqrtL T1=T+F M N=M/L T2=T1*N C C1 MT Miles Per Hour Miles Hours Hours Hours MT Numbers Hours $/Hr $ 5 31.25 70 2.24 2.236 0.6708 2.9108 2500 500 1455.4 325 473005.00 10 22.5 70 3.11 3.162 0.9486 4.059711 2500 250 1014.93 325 329851.53 15 13.75 70 5.09 3.873 1.1619 6.252809 2500 166.67 1042.13 325 338693.83 20 5 70 14 4.472 1.3416 15.3416 2500 125 1917.70 325 623252.50 So we get the minimum cost with truck load L=10MT with truck speed S= 22.5 Miles per Hour , Number of Trips N = 250 & Total Time T2 = 1014.93 Hours The Cost Function Formula will be C1 = C*T2 = C*T1*N=C*(T+F)*(TL/L)=C*(T+F)*(M/L)=C*((D/S)+0.3*SqrtL)*(M/L)=C*((D/(40-1.75*L))+0.3*SqrtL)*(M/L) With 25 Miles per hour and 40 Miles per Hour of Speed the cost will be higher. Now let us see what happens if distance is increase to 40Miles . Distance travelled in one trip will change to 80Miles . Truck Load Speed Distance travelled in one round trip Time required for each round trip SqrtL Time for loading the Truck Total Time Required for 1 trip Total Load to be transfered Total Number of trips required Total Time required to transfer the complete debris Cost of running the fleet Total Cost for removing the debris L S=40-1.75*L D T F= 0.3*SqrtL T1=T+F M N=M/L T2=T1*N C C1 MT Miles Per Hour Miles Hours Hours Hours MT Numbers Hours $/Hr $ 5 31.25 80 2.56 2.236 0.6708 3.2308 2500 500 1615.4 325 525005.00 10 22.5 80 3.56 3.162 0.9486 4.504156 2500 250 1126.04 325 365962.64 15 13.75 80 5.82 3.873 1.1619 6.980082 2500 166.67 1163.35 325 378087.77 20 5 80 16 4.472 1.3416 17.3416 2500 125 2167.70 325 704502.50 If the distance is increased the lowest cost will be with truck load L=10MT Now let us see if the distance is decreased to 25 Miles. Distance travelled in one trip will change to 50 Miles. Truck Load Speed Distance travelled in one round trip Time required for each round trip SqrtL Time for loading the Truck Total Time Required for 1 trip Total Load to be transfered Total Number of trips required Total Time required to transfer the complete debris Cost of running the fleet Total Cost for removing the debris L S=40-1.75*L D T F= 0.3*SqrtL T1=T+F M N=M/L T2=T1*N C C1 MT Miles Per Hour Miles Hours Hours Hours MT Numbers Hours $/Hr $ 5 31.25 50 1.6 2.236 0.6708 2.2708 2500 500 1135.4 325 369005.00 10 22.5 50 2.22 3.162 0.9486 3.170822 2500 250 792.71 325 257629.31 15 13.75 50 3.64 3.873 1.1619 4.798264 2500 166.67 799.71 325 259905.95 20 5 50 10 4.472 1.3416 11.3416 2500 125 1417.70 325 460752.50 If the distance is decreased the lowest cost will be with truck load = 10MT

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