A farmer has 200 acres of land suitable for cultivating crops A, B and C. The cost per acre of cultivating crop A, crop B and crop C is $40, $60 and $80, respectively. The farmer has $12,600 available for land cultivation. Each acre of crop A requires 20 labor-hours, each acre of crop B requires 25 labor-hours and each acre of crop C requires 40 labor-hours. The farmer has a maximum of 5950 labor-hours available. If he wishes to use all his cultivatable land, the entire budget and all the labor available, how many acres of each crop should he plant? [Please can you show me how to Use the Gauss Jordan Elimination method to solve]
Are you able to set up the equations?
a 30 b 20 c 40
To solve this problem using Gauss Jordan Elimination method, we can set up a system of equations based on the given information.
Let's assume x, y, and z as the number of acres to be planted with crop A, crop B, and crop C respectively.
We can write the following equations based on the cost per acre:
40x + 60y + 80z = 12600 -- Equation 1
Similarly, based on the labor-hours required:
20x + 25y + 40z = 5950 -- Equation 2
To solve this system of equations using the Gauss Jordan Elimination method, we need to create an augmented matrix by combining the coefficients of the variables.
The augmented matrix would be:
[40 60 80 | 12600]
[20 25 40 | 5950]
Now, we can perform row operations to transform the matrix into a reduced row-echelon form.
First, let's divide the first row by 40 to make the leading coefficient 1:
[1 1.5 2 | 315]
[20 25 40 | 5950]
Next, let's multiply the first row by 20 and subtract it from the second row:
[1 1.5 2 | 315]
[0 -5 -40 | 3335]
Now, let's divide the second row by -5 to make the leading coefficient 1:
[1 1.5 2 | 315]
[0 1 8 | -667]
Next, let's multiply the second row by -1.5 and add it to the first row:
[1 0 14 | 1050]
[0 1 8 | -667]
Finally, we have our reduced row-echelon form. The solution to the system of equations is:
x = 1050 acres
y = -667 acres
z = 0 acres
Since the number of acres cannot be negative, we can conclude that the farmer should plant 1050 acres of crop A and 0 acres of both crop B and crop C to use all his cultivatable land, the entire budget, and all the available labor.