A farmer has 20 acres to plant in wheat and barley. He has to plant at least 15 acres. Wheat costs $200 per acre to plant and barley costs $150 per acre, and the farmer has only $2400 to spend. If the estimated profit of an acre of wheat is $450 and the estimated profit of an acre of barley is $350, how many acres of each should the farmer plant to maximize his profits?

It is very helpful, all you have to do is plug in the equations into a graph and then plug the intercepts back into the equations. Do some work for yourself, not everyone is going to feed you with a silver spoon.

Just as with the baker, let

x = acres of wheat
y = acres of barley

You want to

maximize p=450x+350y subject to

x+y >= 15
x+y <= 20
200x + 150y <= 2400

Brat

By the way, the answer is 16 barley 0 wheat

Wait but it says for each. so how about for wheat

To maximize the farmer's profit, we need to determine the number of acres of each crop the farmer should plant. Let's assume the farmer plants x acres of wheat and y acres of barley.

We have two constraints:
1. Total acreage planted should be at least 15 acres: x + y ≥ 15
2. The farmer has a total budget of $2400: 200x + 150y ≤ 2400

Next, let's determine the objective function, which is the profit equation. The profit from wheat is $450 per acre, and the profit from barley is $350 per acre. So, the objective function for the farmer's profit is:

Profit = 450x + 350y

Now, we can solve this optimization problem by using linear programming techniques. The first step is to graph the feasible region, which represents all possible combinations of x and y that satisfy the constraints.

To graph the feasible region, plot the lines x + y = 15 and 200x + 150y = 2400 on a coordinate plane. Then, shade the region that satisfies both constraints.

Next, find the coordinates of the corners or vertices of the shaded region. We can do this by solving the system of equations formed by the lines x + y = 15 and 200x + 150y = 2400.

Once we find the coordinates of the vertices, substitute those values into the objective function (Profit = 450x + 350y) to calculate the profit at each vertex.

Finally, compare the profits at each vertex and determine which combination of x and y yields the maximum profit.

Please note that the explanation provided above is the general approach to solve this optimization problem mathematically. However, since the number of acres is relatively small in this case, you can also solve it by trying different combinations and calculating the profits directly.

not helping