A workshop produces two types of wooden figures: soldiers and dancers. Let `x` be the number of soldiers and `y` be the…

First, translate the problem's constraints into a system of linear inequalities. Let `x` be the number of soldiers and `y` be the number of dancers.

1. Carving time constraint: `2x + 3y ≤ 120`

2. Painting time constraint: `x + 2y ≤ 70`

3. Non-negativity constraints: `x ≥ 0` and `y ≥ 0`, since the number of figures cannot be negative.

Next, define the objective function for the total profit, `P`:

`P = 5x + 8y`

The goal is to find the maximum value of `P` for values of `x` and `y` that satisfy all the inequalities. In linear programming, the maximum value of the objective function will occur at one of the vertices of the feasible region defined by the constraints.

To find the vertices, we find the points where the boundary lines of the inequalities intersect.

- Vertex 1: Intersection of `x=0` and `y=0` is `(0, 0)`.

- Vertex 2: Intersection of the painting constraint `x + 2y = 70` with the y-axis (`x=0`). `0 + 2y = 70`, so `y = 35`. The vertex is `(0, 35)`. (Check validity in other constraint: `2(0) + 3(35) = 105 ≤ 120`. Valid.)

- Vertex 3: Intersection of the carving constraint `2x + 3y = 120` with the x-axis (`y=0`). `2x + 0 = 120`, so `x = 60`. The vertex is `(60, 0)`. (Check validity in other constraint: `60 + 2(0) = 60 ≤ 70`. Valid.)

- Vertex 4: Intersection of the two main constraint lines, `2x + 3y = 120` and `x + 2y = 70`. From the second equation, `x = 70 - 2y`. Substitute this into the first equation: `2(70 - 2y) + 3y = 120`. This simplifies to `140 - 4y + 3y = 120`, which gives `140 - y = 120`, so `y = 20`. Substitute `y = 20` back into `x = 70 - 2y` to get `x = 70 - 2(20) = 30`. The vertex is `(30, 20)`.

Now, evaluate the profit `P = 5x + 8y` at each vertex:

- At (0, 0): `P = 5(0) + 8(0) = $0`

- At (0, 35): `P = 5(0) + 8(35) = $280`

- At (60, 0): `P = 5(60) + 8(0) = $300`

- At (30, 20): `P = 5(30) + 8(20) = 150 + 160 = $310`

Comparing the values, the maximum possible daily profit is $310. · *Distractor analysis: · *

A is incorrect. $280 is the profit at the vertex (0, 35). This result comes from maximizing the number of the more profitable item, dancers, but this combination does not yield the maximum overall profit.

B is incorrect. $300 is the profit at the vertex (60, 0). This result comes from maximizing the number of soldiers, but this combination does not yield the maximum overall profit.

D is incorrect. $320 is the profit that would be made at (0, 40). This point is the y-intercept of the carving constraint line (`2x + 3y = 120`), but it is not a valid point because it violates the painting constraint (`0 + 2(40) = 80`, which is not less than or equal to 70). This error arises from considering only one constraint instead of the entire feasible region.