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Question

# How many acres of each (wheat and rye) should the farmer plant in order to get maximum profit?

A
(5,5)
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B
(4,4)
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C
(4,5)
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D
(4,3)
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Solution

## The correct option is A (4,4)let x be the acres of wheat planted andy be the acres of rye plantedGiven that there are a total of 10 acres of land to plant.Atleast 7 acres is to be planted i.e., x+y≥7Given that the cost to plant one acre of wheat is $200Therefore, the cost for x acres of wheat is 200xGiven that the cost to plant one acre of rye is$100Therefore, the cost for y acres of rye is 100yGiven that, amount for planting wheat and rye is $1200Therefore the total cost to plant wheat and rye is 200x+100y≤1200⟹2x+y≤12Given that, the time taken to plant one acre of wheat is 1 hrTherefore, the time taken to plant x acres of wheat is x hrsGiven that, the time taken to plant one acre of rye is 2 hrsTherefore, the time taken to plant y acres of rye is 2y hrsGiven that, the total time for planting is 12 hrsTherefore, the total time to plant wheat and rye is x+2y≤12Given that, one acre of wheat yields a profit of$500Therefore, the profit from x acres of wheat is 500xGiven that, one acre of rye yields a profit of \$300Therefore, the profit from y acres of wheat is 300ytherefore the total profit from the wheat and rye is P=500x+300yNow substituting the options in the profit expression and verifyingSubstituting option A (x,y)=(5,5)x+y≥0⟹5+5≥7⟹10≥7 True2x+y≤12⟹2(5)+5≤12⟹15≤12 FalseSubstituting option B (x,y)=(4,4)x+y≥0⟹4+4≥7⟹8≥7 True2x+y≤12⟹2(4)+4≤12⟹12≤12 Truex+2y≤12⟹4+2(4)≤12⟹12≤12 TrueSubstituting option C (x,y)=(4,5)x+y≥0⟹4+5≥7⟹9≥7 True2x+y≤12⟹2(4)+5≤12⟹13≤12 FalseSubstituting option D (x,y)=(4,3)x+y≥0⟹4+3≥7⟹7≥7 True2x+y≤12⟹2(4)+3≤12⟹11≤12 Truex+2y≤12⟹4+2(3)≤12⟹10≤12 True(4,4),(4,3) satisfies the constraints. Therefore finding the profit for (4,4), P=500x+300y=500(4)+300(4)=3200for (4,3), P=500x+300y=500(4)+300(3)=2900Therefore the maximum profit is attained at (4,4)

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