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# A farmer has 10 acres of land to plant wheat and rye. He has to plant atleast 7 acres. Each acre of wheat costs $200 and each acre of rye costs$100 to plant. He has only $1200 to spend. Moreover, the farmer has to get the planting done in 12 hours and it takes 1 hour to plant an acre of wheat and 2 hours to plant an acre of rye. An acre of wheat yields a profit of$500 and an acre of rye yields a profit of $300.(Take x and y as the acres of wheat and rye planted respectively). What is the maximum profit that the farmer can make? A$2500
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B
$2800 No worries! Weâ€˜ve got your back. Try BYJUâ€˜S free classes today! C$3100
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D
$3200 Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses Open in App Solution ## The correct option is D$3200let 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, an 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+300yIn the above figure, the blue shaded region is the feasible region with three corner points.(4,4),(2,5),(5,2)Now substituting the corner points the profit equation,substituting (4,4)⟹P=500x+300y=500(4)+300(4)=3200substituting (2,5)⟹P=500x+300y=500(2)+300(5)=2500substituting (5,2)⟹P=500x+300y=500(5)+300(2)=3100$3200 is the maximum profit.

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