A factory produces two types of products, A and B. The resources required to produce 1 ton of each product (electricity and oil) and the profit are given in the table. The factory has a maximum of 7kw of electricity and 16kL of oil available. We need to determine how many tons of each product A and B should be produced to maximize profit, and what the maximum profit will be.

1. Problem Description

A factory produces two types of products, A and B. The resources required to produce 1 ton of each product (electricity and oil) and the profit are given in the table. The factory has a maximum of 7kw of electricity and 16kL of oil available. We need to determine how many tons of each product A and B should be produced to maximize profit, and what the maximum profit will be.

2. Solution Steps

Let $x$ be the amount (in tons) of product A produced and $y$ be the amount (in tons) of product B produced.

From the table, the constraints are:

Electricity: $x + y \le 7$

Oil: $x + 4y \le 16$

Also, $x \ge 0$ and $y \ge 0$.

The objective function (profit) to be maximized is: $P = 2x + 3y$.

We need to find the feasible region determined by the constraints. First, let's find the intersection points of the boundary lines:

$x + y = 7$ and $x + 4y = 16$

Subtracting the first equation from the second gives $3y = 9$, so $y = 3$.

Substituting $y = 3$ into $x + y = 7$ gives $x + 3 = 7$, so $x = 4$.

Thus, the intersection point is $(4, 3)$.

The corner points of the feasible region are:

$(0, 0)$

$(7, 0)$

$(0, 4)$

$(4, 3)$

Evaluate the objective function at each corner point:

At $(0, 0)$: $P = 2(0) + 3(0) = 0$

At $(7, 0)$: $P = 2(7) + 3(0) = 14$

At $(0, 4)$: $P = 2(0) + 3(4) = 12$

At $(4, 3)$: $P = 2(4) + 3(3) = 8 + 9 = 17$

The maximum profit occurs at $(4, 3)$, with a profit of

1

7.

3. Final Answer

To maximize profit, the factory should produce 4 tons of product A and 3 tons of product B. The maximum profit will be 17 (万円).