Answer:
The number of acre per crop for maximum profit is;
Crop A = 0 acres
Crop B = 60 acres
Crop C = 40 acres
Profit, P = $26,000
Step-by-step explanation:
We plant A in X acres
B in Y acres and
C in Z acres
Therefore, X + Y + Z ≤ 100
we work A for 1·X workdays
B for 2·Y workdays and
C for 1·Z workdays
Where X + 2·Y + Z ≤ 160
$40·X for crop A
$20·Y for crop B and
$30·Z for crop C is spent whereby
$40·X + $20·Y + $30·Z ≤ $3200
The farmer makes
$100·X from crop A
$300·Y from crop B and
$200·Z from crop C
P = $100·X + $300·Y + $200·Z
Therefore, we have three equations with three unknowns solving the equations simultaneously, we have
X + Y + Z = 100....................................................(1)
X + 2·Y + Z = 160.................................................(2)
$40·X + $20·Y + $30·Z = $3200....................(3)
By subtracting equation (1) from (2) gives Y = 60 acres
Multiplying equation (1) by 40 and subtracting from (3) we have Z = -40
and therefore, Y = 80
If he plants only B the farmer has 2 work day per acre and since there is a max of 160 days, he can only plant on 80 acres, therefore, total profit = $300 × 80 = $24,000
Comparing the profit per acre to the seed cost we have Profit for seed A = $100 while cost = $40 per acre,
If we remove seed A we have
Y + Z = 100....................................................(4)
2·Y + Z = 160.................................................(5)
$40·X + $20·Y + $30·Z = $3200....................(3)
Solving equation (4) and (5), we have Y = 60 acres and Z = 40 acres
Therefore, the profit becomes
$300 × 60 + $200 × 40 = $26,000.
