Let's solve this optimization problem using a linear programming approach.

Variables

Let:

• $x$ = the number of acres planted with crop A.
• $y$ = the number of acres planted with crop B.

Objective

The goal is to maximize profit, given by: $\text{Profit} = 30x + 90y$

Constraints

1. Total Acreage Constraint: The farmer has a total of 50 acres. $x + y \leq 50$

2. Seed Cost Constraint: The farmer has a budget of $1400 for seeds. Since crop A costs $40 per acre and crop B costs $20 per acre: $40x + 20y \leq 1400$

3. Non-negativity Constraints: The number of acres planted for each crop must be non-negative: $x \geq 0 \quad \text{and} \quad y \geq 0$

Solving the Linear Programming Problem

1. Rewrite the Seed Cost Constraint: Simplify the second inequality by dividing by 20: $2x + y \leq 70$

So, our problem becomes:

Maximize: $30x + 90y$

Subject to: $x + y \leq 50$ $2x + y \leq 70$ $x \geq 0, \quad y \geq 0$

Finding the Feasible Region and Vertices

To maximize profit, we need to check the vertices of the feasible region formed by these inequalities. Let's find the points of intersection.

1. Intersection of $x + y = 50$ and $2x + y = 70$

Solve these equations simultaneously:

1. From $x + y = 50$, we get $y = 50 - x$.
2. Substitute $y = 50 - x$ into $2x + y = 70$: $2x + (50 - x) = 70$ $x + 50 = 70$ $x = 20$
3. Substitute $x = 20$ back into $y = 50 - x$: $y = 50 - 20 = 30$

So, one vertex is $(20, 30)$.

2. Intersection of $x + y = 50$ and $y = 0$

If $y = 0$: $x + 0 = 50 \Rightarrow x = 50$ So, another vertex is $(50, 0)$.

3. Intersection of $2x + y = 70$ and $y = 0$

If $y = 0$: $2x + 0 = 70 \Rightarrow x = 35$ So, another vertex is $(35, 0)$.

Now we have three vertices: $(20, 30)$, $(50, 0)$, and $(35, 0)$.

Evaluating the Objective Function at Each Vertex

Let's calculate the profit at each of these points:

1. At $(20, 30)$: $\text{Profit} = 30(20) + 90(30) = 600 + 2700 = 3300$

2. At $(50, 0)$: $\text{Profit} = 30(50) + 90(0) = 1500$

3. At $(35, 0)$: $\text{Profit} = 30(35) + 90(0) = 1050$

Conclusion

The maximum profit is achieved at the point $(20, 30)$, where the profit is $3300.

Answer

The farmer should plant:

• 20 acres of crop A
• 30 acres of crop B

This configuration maximizes her profit to $3300.

Would you like more details or have any questions?

Related Questions

1. How would the solution change if the cost of crop A's seed increased to $50 per acre?
2. What happens to the profit if the profit per acre of crop B decreases?
3. How would the solution be affected if the total available land was reduced to 40 acres?
4. Can we add another crop, say crop C, and solve the optimization with three crops?
5. What is the impact on profit if the farmer spends exactly $1400 on seeds?

Tip:

In linear programming, identifying the vertices of the feasible region is crucial, as the maximum or minimum values of the objective function always occur at these points.