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A Farmer Is Planning His Spring Planting Suppose the Farmer Can Purchase More Fertilizer for $2

question 44

Essay

A farmer is planning his spring planting.He has 20 acres on which he can plant a combination of Corn,Pumpkins and Beans.He wants to maximize his profit but there is a limited demand for each crop.Each crop also requires fertilizer and irrigation water which are in short supply.The following table summarizes the data for the problem.
 Crop  Profit per  Acre $) Yield per  Acre 1b)  Maximum  Demønd lb)  Irrigation  acre ft)  Fertilizer  poundls/acre)  Corn 2,10021,000200,0002500 Pumplin 90010,000180,0003400 Beans 1,0503,50080,0001300\begin{array}{r|rcccc}\text { Crop } & \begin{array}{c}\text { Profit per } \\\text { Acre } \$)\end{array} & \begin{array}{c}\text { Yield per } \\\text { Acre 1b) }\end{array} & \begin{array}{c}\text { Maximum } \\\text { Demønd lb) }\end{array} & \begin{array}{c}\text { Irrigation } \\\text { acre ft) }\end{array} & \begin{array}{c}\text { Fertilizer } \\\text { poundls/acre) }\end{array} \\\hline \text { Corn } & 2,100 & 21,000 & 200,000 & 2 & 500 \\\text { Pumplin } & 900 & 10,000 & 180,000 & 3 & 400 \\\text { Beans } & 1,050 & 3,500 & 80,000 & 1 & 300\end{array}

Suppose the farmer can purchase more fertilizer for $2.50 per pound,should he purchase it and how much can he buy and still be sure of the value of the additional fertilizer? Base your response on the following Analytic Solver Platform sensitivity output.
changing cills\text {changing cills}
 Cell  Name  Final  Value  Reduced  Cost  Objective  Coefficient  Allowable  Increase  Allowable  Decrease $ B$4 Acres of Corn 9.52021001E+30350$C$4 Acres of Pumpkin 0500.01899.99500.011E+30$D$4 Acres of Beans 10.7901050210375.00\begin{array}{rrrrrrr}\text { Cell } & \text { Name } & \begin{array}{r}\text { Final } \\\text { Value }\end{array} & \begin{array}{r}\text { Reduced } \\\text { Cost }\end{array} & \begin{array}{r}\text { Objective } \\\text { Coefficient }\end{array} & \begin{array}{r}\text { Allowable } \\\text { Increase }\end{array} & \begin{array}{r}\text { Allowable } \\\text { Decrease }\end{array} \\\hline \$ \mathrm{~B} \$ 4 & \text { Acres of Corn } & 9.52 & 0 & 2100 & 1 \mathrm{E}+30 & 350 \\\$ \mathrm{C} \$ 4 & \text { Acres of Pumpkin } & 0 & -500.01 & 899.99 & 500.01 & 1 \mathrm{E}+30 \\\$ \mathrm{D} \$ 4 & \text { Acres of Beans } & 10.79 & 0 & 1050 & 210 & 375.00\end{array}

 Constraints \text { Constraints }
 Cell  Name  Final  Value  Shadow  Price  Constraint  R.H. Side  Allowable  Increase  Allowable  Decrease $E$8 Com demand Used 2000000.017200000136000152000$E$9 Pumplin demand Used 001800001E+30180000$E$10 Bean demand Used 37777.780800001E+3042222.22$E$11 Water Used 29.840501E+3020.15 $F$12  Fertilizer IJsed 80003580003619.043238.09\begin{array}{ccccccc} \text { Cell } & \text { Name } & \begin{array}{l}\text { Final } \\\text { Value }\end{array} & \begin{array}{r}\text { Shadow } \\\text { Price }\end{array} & \begin{array}{c}\text { Constraint } \\\text { R.H. Side }\end{array} & \begin{array}{r}\text { Allowable } \\\text { Increase }\end{array} & \begin{array}{r}\text { Allowable } \\\text { Decrease }\end{array} \\\hline \$ E \$ 8 & \text { Com demand Used } & 200000 & 0.017 & 200000 & 136000 & 152000 \\\$ E \$ 9 & \text { Pumplin demand Used } & 0 & 0 & 180000 & 1 E+30 & 180000 \\ \$ E \$ 10 & \text { Bean demand Used } & 37777.78 & 0 & 80000 & 1 \mathrm{E}+30 & 42222.22 \\ \$ E \$ 11 & \text { Water Used } & 29.84 & 0 & 50 & 1 E+30 & 20.15 \\\text { \$F\$12 } & \text { Fertilizer IJsed } & 8000 & 35 & 8000 & 3619.04 & 3238.09\\ \end{array}


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