Agriculture A farmer has 1200 acres of land on which he grows corn, wheat, and soybeans. It cost

Agriculture A farmer has 1200 acres of land on which he...

Key Concepts

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. In allocation and budgeting problems, these equations model the relationships among different variables, such as quantities produced and costs incurred, providing a mathematical framework to determine values that satisfy all constraints simultaneously.

Constraint Equations

Constraint equations are used to represent the limitations or requirements in a problem, such as resource limits, budgets, or proportional relationships between variables. They dictate the feasible region of solutions and ensure that any solution complies with the given restrictions or conditions.

Substitution Method

The substitution method involves solving one equation for a variable and then substituting that expression into another equation. This technique simplifies a system of equations by reducing the number of variables, making it easier to find a solution that satisfies all the constraints.

Resource Allocation Modeling

Resource allocation modeling involves distributing limited resources among various competing activities through mathematical formulations. It is crucial in operations research and economics for making decisions on how best to utilize resources under given constraints like total availability and specific cost conditions.

Transcript

00:01 Given question we have a farmer who has 200 acres of land agricultural land and is using this 200 acres of agriest land to grow three types of crops one is corn second is wheat and soabin so the three types of crops are corn wheat and the third type of crop is so i've been so here we have also given another information that he has allocated 63 ,750 for the cost of growing his crops and the cost of growing corn is forty five dollars per acre and it is sixty dollars per acre for wheat and for soabins it is forty five so the cost for corn is $45 and cost for wheat is $60 and cost for soya is $50.

01:40 And net cost that the farmer can bear is $63 ,750.

01:56 Now here we don't know the amount of land allocated.

02:06 For each of the crops now we need to calculate the amount of land that has been distributed out of the 200 acres of land in total to each of the crops grown here we have also been told that because of market demand he is growing twice as many acres of wheat as of corn so the amount of land allocated for corn be x acres amount of land allocated for wheat is 2x acres from the conditions given and the amount of land allocated for soya be y acres the total area allocated is or obviously limited to 200 acres now we have to formulate linear equations from these given conditions the first equation that we obtain is x plus 2x plus y is equal to 1200 so therefore 3x plus y is equal to 1200 so this is the first equation that we obtain now from the cost of growing crops multiplied with the land because this is cost per acre so we have x acres so 45x for corn plus 16 to 2 x which is 120x for wheat plus 50 into y dollars for soya the total cost should be limited to 63 ,075 dollars so this is our equation number two if we add 124x that's 50 y is equal to 63 ,000 7550 now yeah so here sorry here we have 45x right this is 45 so 45 plus 120 comes out to be 165 so this is 165 x this is our equation number 2 solving equation 1 and 2 will get our answer so to do that we will multiply 10 times equation 1 and subtract it from equation 2 so equation 1 into 10 to eliminate y here minus equation 2 so by doing this we get 30x plus 10 y is equal to 12 ,000 and in equest 2 and in equations into we have 165x plus 50 y is equal to 63 750 so subtracting this no no we have to multiply 50 here 50 times equation 1 so this becomes 150 150 150 x and here it becomes 50 into 1200 is 60 thousand now subtracting these two in equations we get this is 50 here 15 x is equal to 3 ,075 so x is equal to 250 acres now putting the value of x in equation 1 we get three times of 250 plus y is equal to 1 ,200 now this implies y is equal to to 1200 minus 750 which is 450 so this is value of 5 here and so therefore the area allocated for corn will be 250 acres for wheat it will be 500 acres and for soyabine it will be 450 acres and for soya bean it will be 450 acres so these are also listen so writing it involves.

10:39 So for corn, the area is allocated is 250 acres...

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