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).
