In order to grow a certain crop, it is recommended that each square foot of the...

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In order to grow a certain crop, it is recommended that eachsquare foot of the ground be treated with 10 units of phosphorus, 9units of potassium and 19 units of nitrogen. Suppose that there arethree brands of fertilizer on the market - X, Y and Z. One pound ofX contains 2, 3, 5 units of phosphorus, potassium and nitrogenrespectively. One pound of Y contains 1, 3, 4 units of phosphorus,potassium and nitrogen respectively. One pound of Z contains 1, 0,1 units of phosphorus, potassium and nitrogen respectively. Usingmatrixes:
(i) Determine whether or not it is possible to meet exactly therecommendations by applying some combination of the three brands offertilizer?
(ii) Take into account that a negative number of pounds of anybrand cannot be applied, and suppose that each brand is sold onlyin integral amounts. Under these conditions, determine all possiblecombinations of the three brands that can be applied to satisfy therecommendations exactly.
(iii) Suppose X costs $1 per pound, Y costs $6 per pound and Zcosts $3 per pound. Determine the least expensive solution thatwill satisfy the recommendations exactly as well as the constraintsin (ii).

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4.2 Ratings (763 Votes)

10 units of phosphorus 9 units of potassium and 19 units ofnitrogen are needed Now fertilizer X has 235 units of the threechemicals fertilizer Y has 134 units of the chemicals andfertilizer Z contains 101 units Let x yz denote the amounts of each fertilizer purchasedThen to meet these requirements exactly we get the followingsystem of equationsLet us see if we can solve this system of equations See Answer

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