A farmer has 94 acres on which to plant oats or corn. Each acre of oats requires

Question

A farmer has 94 acres on which to plant oats or corn. Each acre of oats requires $18 capital and 2 hours of labor. Each acre of corn requires $36 capital and 6 hours of labor. Labor costs are $8 per hour. The farmer has $2100 available for capital and $2400 available for labor. If the revenue is $54 from each acre of oats and $102 from each acre of corn, what planting combination will produce the greatest total profit? (Profit here is revenue plus leftover capital and labor cash reserve.) What is the maximum profit?

Explanation / Answer

Requirement for corn:

$36 capital and 6 hrs of labor

Requirement for oats:

$18 capital and 2 hrs of labor

Total land = 94 acre

Labor cost = $8 per hr

Revenue from oat per acre = $54

Revenue from corn per acre = $102

Available money for capital = $2100

Available money for labor = $2400

Constraints will be:

oats + Corn <= 94 acre

Revenue:

Maximize(54*oats + 102*corn)

Capital

18o + 36c <= 2100

3o + 6c <= 350

labor

2*8o + 6*8c <=2400

16o + 48c <= 2400

2o + 6c <= 300

Now:

Final Equations are:

1. o + c <= 100

2. 3o + 6c <= 350

3. 2o + 6c <= 300

4. maximize (54*o + 102*c)

Solving these constraints will give

land for oat = 214/3 acre

land for corn = 68/3 acre

cost = 214*18/3 + 214*2*8/3 + 68*36/3 + 38*6*8/3 = $3849.33

Revenue = 54*214/3 + 102*68/3 = $6164

Profit = 6164 - 3849.33 = $2314.67

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