Task 2 Your company is preparing a site for construction by excavating and remov

Question

Task 2 Your company is preparing a site for construction by excavating and removing earth. there is 420 cubic metres of earth to be removed off a block of block of land. You have two types of trucks available: trucks of 6 cubic metres that cost $60 per load to operate and trucks of 8 cubic metres that cost $70 per load. The smaller truck takes 6 mins to load and the larger truck takes 10 mins. The excavator must use both trucks but can load only one truck at a time and will stop operations after 8 hours. Work out what combination of trucks that will shift the earth in 8 hours for the smallest cost.

Explanation / Answer

Let us assume that x number of smaller trucks of 6 cu. meters and y number of bigger trucks of 8 cu. meters is used. Since the quantity of earth required to be moved is 420 cu. meters, hence 6x + 8y = 420 …(1)

Further, since the smaller trucks take 6 minutes to fill and the larger trucks take 10 minutes to fill, and since a total time slot of 8 hours ( = 8 * 60 = 480 minutes) is available, we have 6x + 10y = 480…(2). Now, on subtracting the 1st equation from the 2nd equation, we have 6x + 10 y – 6x – 8y = 480 – 420 or, 2y = 60 so that y = 60/2 = 30. On substituting this value of y in the 1st equation, we have 6x + 8*30 = 420 or, 6x + 240 = 420 or, 6x = 420 -240 = 180 so that x = 180/6 = 30.

Thus 30 smaller trucks and 30 bigger trucks would remove 420 cu. meters of earth in 8 hours at a cost of 30*60 + 30*70 = 1800 + 2100 = $ 3900. Prima-facie, it appears that the cost of operating the smaller truck is lower. However, the cost of removing 1 cu. meter of earth, if we use a smaller truck is $ 60/6 or, $ 10 and the cost of removing 1 cu. meter of earth, if we use a bigger truck is $ 70/8 = $ 8.75. Thus, it is the bigger trucks which are more cost efficient. Let us check whether using only the bigger trucks will reduce cost. To remove 420 cu. meters of earth, we will require 420/8 = 52.5 or, say 53 bigger trucks. The cost would, then be 53*70 = $ 3710 which is lower than $ 3900. However, the time taken will be 53*10 = 530 minutes = 530/60 = 8 hours, 50 minutes which is not available. Hence 30 smaller trucks and 30 bigger trucks will do the job in a cost efficient manner within the given time constraints.

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