Respuesta :
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.
The 0 acres of crop A, 60 acres of crop B, and 40 acres of crop c each crop should be planted to maximize profit $26,000.
Given that,
A farmer owns a 100-acre farm and plans to plant at most three crops. The seed for crops A, B, and C costs $40, $20, and $30 per acre, respectively.
A maximum of $3200 can be spent on the seed. Crops A, B, and C require 1,2, and 1 workday per acre, respectively, and there are a maximum of 160 workdays available.
If the farmer can make a profit of $100 per acre on crop A, $300 per acre on crop B, and $200 per acre on crop C,
We have to find,
How many acres of each crop should be planted to maximize profit?
According to the question,
The farmer can make a profit of $100 per acre on crop A, $300 per acre on crop B, and $200 per acre on crop C,
[tex]\rm P = 100A + 300B + 200C[/tex]
The seed for crops A, B, and C costs $40, $20, and $30 per acre, respectively.
And A maximum of $3200 can be spent on the seed.
[tex]\rm 40A + 20B + 30C \leq 3200[/tex]
Then, Sum of seed for crops costs = area of the farm
[tex]\rm A + B + C \leq 100[/tex]
Crops A, B, and C require 1,2, and 1 workday per acre, respectively, and there are a maximum of 160 workdays available.
[tex]\rm A + 2B + C = 160[/tex]
Solving all the equations,
From equation 3,
[tex]\rm A = 100-B-C[/tex]
Substitute the value of A in equation 4,
[tex]\rm 100 - B - C+ 2B + C = 160\\\\100 + B = 160 \\\\B = 160 - 100\\\\B = 60[/tex]
Put B = 60 in the equation,
[tex]\rm A = 100-B-C\\\\A = 100-60-C\\\\A = 40-C[/tex]
Substitute the value of A in equation 2,
[tex]\rm 40(40-C) + 20\times 60 + 30C = 3200\\\\1600 - 40C + 1200 + 30C = 3200\\\\-10C + 2800 = 3200\\\\-10C = 3200-2800\\\\-10C = 400\\\\C = \dfrac{400}{-10}\\\\C = -40[/tex]
The area can not be negative then the value of C is +40.
And the value of A is,
[tex]A + B + C = 100\\\\A = 100 - 60 -40\\\\A = 100-100\\\\A = 0[/tex]
Then, Maximize profit is,
[tex]\rm P = 100A + 300B + 200C\\\\\rm P = 100(0) + 300(60) + 200(40)\\\\P =0+18000+8000\\\\P = 26000[/tex]
Hence, The 0 acres of crop A, 60 acres of crop B, and 40 acres of crop c each crop should be planted to maximize profit $26,000.
For more details refer to the link.
https://brainly.com/question/19712040
