A chemical manufacturer wants to lease a fleet of 22 railroad tank cars with a c

Question

A chemical manufacturer wants to lease a fleet of 22 railroad tank cars with a combined carrying capacity of 312,000 gallons. Tank cars with three different carrying capacities are available: 6,000 gallons, 12,000 gallons, and 24,000 gallons. Letting t = number of 24,000 railroad tank cars in the fleet, 2-8 6,000 gallon railroad tank cars, and 3t+30 12,000 gallon railroad tank cars will be in the fleet for a certain range of t. The cost of leasing a 6,000-gallon tank car is $350 per month, a 12,000-gallon tank car is $450 per month, and a 24,000-gallon tank car is $1050 per month. Which of the solutions to the number of each type of tank car in the fleet would minimize the monthly leasing cost? 6,000-gallon tank cars, 12,000-gallon tank cars, and [ (Simplify your answers. Type whole numbers.) 24,000-gallon tank cars.

Explanation / Answer

The cost of leasing a 24000 gallons tank car is $ 1050 per month i.e. $ 43.75 per 1000 gallons, the cost of leasing a 12000 gallons tank car is $ 450 per month i.e. $ 37.50 per 1000 gallons, and the cost of leasing a 6000 gallons tank car is $ 350 per month i.e. $ 58.33 per 1000 gallons. Thus, the monthly leasing charges are the least for 12000 gallons tank cars and maximum for 6000 gallons tank cars. Hence the total cost of leasing would be minimized if 2t-8 is minimized and -3t+30 is maximized. The minimum value of 2t -8 is 0. Then t = 4 and –3t+30 = 18.   Also, 0*6000 +18*12000 +4*24000 = 312000

Thus,the total cost of leasing would be minimized if 0- 6000 gallons tank cars, 18-12000 gallons tank cars and 4 -24000 gallons tank cars are leased.     

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