A grain company wants to lease a fleet of 20 covered hopperrailcars witha combin

Question

A grain company wants to lease a fleet of 20 covered hopperrailcars witha combined capacity of 108000ft^3. Hoppers with threedifferent carrying capacities are available: 3000ft^3, 4500ft^3 and6000ft^3. a) How many of each type of hopper should they lease? (thereare no fractional hoppers!) b) The monthly leasing rates are $180, 225 and 325 for the3000ft^3, 4500ft^3 and 6000ft^3 hoppers respectively. Whichsolution in a) minimizes the monthly leasing? A grain company wants to lease a fleet of 20 covered hopperrailcars witha combined capacity of 108000ft^3. Hoppers with threedifferent carrying capacities are available: 3000ft^3, 4500ft^3 and6000ft^3. a) How many of each type of hopper should they lease? (thereare no fractional hoppers!) b) The monthly leasing rates are $180, 225 and 325 for the3000ft^3, 4500ft^3 and 6000ft^3 hoppers respectively. Whichsolution in a) minimizes the monthly leasing?

Explanation / Answer

a) Let x be the number of 3000ft hoppers       y be the number of 4500fthoppers       z be the number of 6000fthoppers Then: 3000x + 4500y + 6000z = 108000   (1) and x + y + z = 20      (2) Now, we can say x = 20 - z - y and sub this into (1) to give 3000(20 - z - y) + 4500y + 6000z = 10800 Therefore (4500 - 3000)y + (6000 - 3000)z = 108000 - 3000*20 hence 1500y + 3000z = 48000 15y + 30z = 480 y + 2z = 32 Therefore y = 32 - 2z Trial and error y = 32 - 2 = 30 X y = 32 - 2*2 = 28 X ... ... y = 32 - 2*6 = 20   ... (y,z,x) = (20,6, ... X y = 32 - 2*7 = 18   ... (y,z,x) = (18.7 ...X y = 32 - 2*8 = 16 X y = 32 - 2*9 = 14 X y = 32 - 2*10 = 12 X y = 32 - 2*11 = 10 X y = 32 - 2*12 = 8 ... (y,z,x) =(8, 12, 0) y = 32 - 2*13 = 6 ... (y,z,x) =(6,13,1) y = 32 - 2*14 = 4 ... (y,z,x)  =(4,14,2) y = 32 - 2*16 = 2 ...(2,16,2) y = 32 - 2*18 = 0 ... (0,18,2) Subbing these into equation (1) you will find the solutionsthat work. Should help you to pick one for b Hope that helps

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