Algebra.Com's Answer #812435 by ikleyn(52905)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "For the anniversary celebration, there will be helium-filled balloons for the children.
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document.write( "They have purchased spherical balloons that have a 9 in. diameter.
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document.write( "When filled with helium, the balloons should continue to fly for most of the day.
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document.write( "The committee estimates that they will need 1 500 balloons.
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document.write( "Part 1:
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document.write( "They plan to purchase the cylinders of helium and fill the balloons themselves.
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document.write( "They have two choices of cylinders:
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document.write( "a medium cylinder which contains 80 cu. ft. of helium
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document.write( "a large cylinder which contains 100 cu. ft. of helium
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document.write( "The committee does not want to purchase too much extra helium, but they need to have enough to fill all 1500 balloons.
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document.write( "How many balloons can be filled using a medium cylinder?
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document.write( "How many balloons can be filled using a large cylinder?
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document.write( "How much helium is needed to fill all the balloons?
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document.write( "Advise the committee of which cylinder(s) to purchase and how many of each they should order. (Ignore cost.)
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document.write( "One balloon volume =  =  = 381.7032 in^3 =  = 0.220893047 ft^3.\r\n" );
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document.write( "1500 ballons volume = 1500*0.220893047 ft^3 = 331.3395703 ft^3 - total necessary volume of helium for 1500 balloons.\r\n" );
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document.write( "The further part of the problem is posed uncertainly; but we can interpret it as the requirement to have the total volume \r\n" );
document.write( "of the helium above 331.33957 ft^3, but minimize the extra helium.\r\n" );
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document.write( "More formally, we should have\r\n" );
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document.write( " 80x + 100y >= 331.33957 ft^3 of helium\r\n" );
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document.write( " x >= 0, y >= 0, where x and y are integer numbers;\r\n" );
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document.write( " 80x + 100y - 331.33957 ---> minimum.\r\n" );
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document.write( "The simplest way to solve this minimization problem is to make a table of reasonable values of x and y \r\n" );
document.write( "(the numbers of cylindrical containers of both types) and determine the necessary minimum from the table.\r\n" );
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document.write( " T A B L E\r\n" );
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document.write( " x y 80x + 100y the extra helium volume\r\n" );
document.write( " 80x + 100y - 331.33957\r\n" );
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document.write( " 5 0 400 68.66043\r\n" );
document.write( " 4 1 420 88.66043\r\n" );
document.write( " 3 1 340 8.66043\r\n" );
document.write( " 2 2 360 28.66043\r\n" );
document.write( " 1 3 380 48.66043\r\n" );
document.write( " 0 4 400 68.66043\r\n" );
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document.write( "The numbers in the first two column are chosen in this way to cover all reasonable options\r\n" );
document.write( "and do not consider (cut off) all UNREASONABLE versions.\r\n" );
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document.write( "The calculations show that all these options provide the necessary volume of the helium\r\n" );
document.write( "and the minimum of extra helium is at x = 3 (three 80 ft^3 helium containers) and y = 1 (one 100 ft^3 helium container).\r\n" );
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document.write( "ANSWER. Among all possible combinations of the helium containers, that provide\r\n" );
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document.write( " the necessary minimum helium volume and minimize extra volume of helium, \r\n" );
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document.write( " the solution is three 80 ft^3 helium containers and one 100 ft^3 helium container.\r\n" );
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document.write( "Solved.\r
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document.write( "It is how I would solve it as a Math problem; how I would present the arguments \r
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document.write( "and how I'd made the final presentation for the higher authority to consider and to approve the project.\r
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