A farmer has to decide how many acres of a 30-acre plot are to be devoted to growing strawberries and how many to growing tomatoes. There will be 220 hours of labor available for the picking. It takes 8 hours to pick an acre of strawberries and 6 hours to pick an acre of tomatoes. The profit per acre is $700 on the strawberries compared to $600 on the tomatoes. How many acres of each should be grown to maximize profit?

Question

A farmer has to decide how many acres of a 30-acre plot are to be devoted to growing strawberries and how many to growing tomatoes. There will be 220 hours of labor available for the picking. It takes 8 hours to pick an acre of strawberries and 6 hours to pick an acre of tomatoes. The profit per acre is $700 on the strawberries compared to $600 on the tomatoes. How many acres of each should be grown to maximize profit?

Answer 1

constraint: S+T<=30

constraint: 8S+6T<=220
so, where do these lines intersect?
6S+6T=180
8S+6T=220
or
2S=40 ;S=20 and T=10
so the test points the boundary points (O,30)(220/8,0)(20,10)
so figure profit for each point
P=700S+600T
P1=18,000
P2=19250
P3=20,000
check my work

Answer 2

To find out how many acres of strawberries and tomatoes should be grown to maximize profit, we can use linear programming. In this case, we need to define variables, set up constraints, and formulate the objective function.

Let's assume 'x' represents the number of acres of strawberries, and 'y' represents the number of acres of tomatoes.

Objective Function:
The objective is to maximize profit, so the objective function is:
Profit = 700x + 600y

Constraints:
1. The total number of acres cannot exceed the 30-acre plot, so the first constraint is:
x + y ≤ 30

2. The total number of labor hours cannot exceed the available 220 hours, so the second constraint is:
8x + 6y ≤ 220

3. The number of acres cannot be negative, so the third constraint is:
x ≥ 0, y ≥ 0

Now that we have formulated the problem, we can use a linear programming solver or graphical method to find the optimal solution.

Alternatively, we can solve this problem using the graphical method by plotting the constraints on a graph and finding the feasible region. The overlapping area of the feasible region will give us the optimal solution.

Steps for graphical method:
1. Plot the constraints on a coordinate plane.
2. Shade the region that satisfies all the constraints (feasible region).
3. Find the corner points of the feasible region.
4. Evaluate the objective function at each corner point.
5. The corner point with the maximum objective function value will give us the optimal solution.

Using this approach, we can find the optimal solution for the number of acres of strawberries and tomatoes that will maximize profit.