During a war allies sent food and medical kits to help survi

During a war, allies sent food and medical kits to help survivors. Each food kit helped

88

people and each medicine kit helped

44

people. Each plane could carry no more than

50 comma 00050,000

pounds. Each food kit weighed 20 pounds and each medicine kit weighed 10 pounds. In addition to the weight constraint on its cargo, each plane could carry a total volume of supplies that did not exceed

40004000

cubic feet. Each food kit was 1 cubic foot and each medical kit also had a volume of 1 cubic foot. Assume that those helped by medicine kits were not helped by the food kits and vice versa. What was the maximum number of people that could be helped with one plane of supplies?

The maximum number of people that could be helped was ___ people

Solution

Let the number of food kits and the medical kits carried by the plane be 1000x and 1000y respectively. Since each plane could carry no more than 50,00050,000 pounds and since each food kit weighed 20 pounds and each medicine kit weighed 10 pounds, we have 20000x+10000y 50,00050,000. On dividing both the sides by 10000, we get 2x+y 500005…(1)

Also, since the volume of each food kit was 1 cubic foot and each medical kit also had a volume of 1 cubic foot and since each plane could carry a total volume of supplies that did not exceed 40004000cubic feet, therefore, we have 1000x+1000y 40004000. On dividing both the sides by 1000, we get x+y 40004…(2)

The feasible region is on or below the lines 2x+y = 500005…(3) and x+y = 40004…(4). On solving these equations, we get 2x+y –(x+y) = 500005-40004 or, x = 460001. On substituting this value of x in the 4^th equation, we get 460001 +y = 40004 or, y = -419997.

This is not possible as both x and y have to be positive. Thus, there seems to be an error in the given data. After correcting the data, we will get the correct values of x and y at the point of intersection of the lines (3) and (4). This will satisfy both these equations. Then the answer is (1000x)88 +(1000y)44 = 88000x+44000y.