Question 1111817: A farmer has 300 square metres of land suitable for cultivating the crops baigan, bodi, and pumpkin. The cost per square metre of cultivating baigan, bodi, and pumpkin is $20, $60, and $70, respectively. The farmer has $13,350 available for land cultivation. Each square metre of baigan requires 10 hours of labour time, each square metre of bodi requires 35 hours, and each square metre of pumpkin requires 30 hours. The farmer has a maximum of 6525 hours of labour time available. If he desires to plant his entire plot of land, use the entire budget, and all the labour time available:
a. Derive a system of three equations in x, y and z, where x is the number of square metres of baigan, y is the number of square metres of bodi, and z is the number of square metres of pumpkin planted.
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b. Use the inverse method to derive, how many square metres of each crop should he plant?
Answer by mananth(16946)   (Show Source):
You can put this solution on YOUR website! Total land = 300 square metres
The cost per square meter of cultivating
x m^2 baigan, =$20 time 10 hours
y m^2 bodi,=$60 , 35 hours,
z m^2 pumpkin $70 ,30 hours.
a maximum of 6525 hours of labour time
Total money = $13,350
10x+35y+30z=6525
20x+60y+70z=13350
x+y+z=300
Your matrix
X1 X2 X3 b
1 1 1 1 300
2 10 35 30 6525
3 20 60 70 13350
main matrix
X1 X2 X3
1 1 1 1
2 10 35 30
3 20 60 70
Determinant is not zero, therefore inverse matrix exists
Calculate the inverse matrix
X1 X2 X3
1 13/9 -1/45 -1/90
2 -2/9 1/9 -2/45
3 -2/9 -4/45 1/18
Multiply the inverse matrix by the solution vector
X
1 140
2 65
3 95
Solution set:
x1 = 140
x2 = 65
x3 = 95
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