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A manufacturer wants to produce x balls and y boxes. Resource constraints require that x and y satisfy the inequalities shown. What is the maximum number of balls and boxes combined that can be produced given the resource constraints?
Let's understand what we're looking for here. We have a manufacturer who wants to make balls \(\mathrm{(x)}\) and boxes \(\mathrm{(y)}\), but they're limited by their resources. Think of it like having a limited budget and limited workers - you can't make unlimited products.
Our goal is to find the maximum total number of items (balls + boxes) they can produce. So we want to maximize: \(\mathrm{x + y}\)
But we have two resource limitations:
Also, since we can't make negative balls or boxes: \(\mathrm{x \geq 0}\) and \(\mathrm{y \geq 0}\)
Process Skill: TRANSLATE - Converting the business problem into mathematical optimization
Now, here's a key insight: the maximum value will occur at a "corner" of our feasible region. Think of it like this - if you're trying to get as far northeast as possible on a map, but you're blocked by walls, you'll end up pressed against a corner where two walls meet.
Let's find these "corners" by looking at where our constraint lines intersect:
Corner 1: Where both constraints are tight (intersection point)
Set both constraints as equalities:
Corner 2: First constraint meets y-axis \(\mathrm{(x = 0)}\)
\(\mathrm{7(0) + 6y = 38,000 \rightarrow y = \frac{38,000}{6} \approx 6,333}\)
Corner 3: Second constraint meets y-axis \(\mathrm{(x = 0)}\)
\(\mathrm{4(0) + 5y = 28,000 \rightarrow y = 5,600}\)
Corner 4: Both constraints meet x-axis \(\mathrm{(y = 0)}\)
First: \(\mathrm{7x = 38,000 \rightarrow x \approx 5,429}\)
Second: \(\mathrm{4x = 28,000 \rightarrow x = 7,000}\)
Process Skill: VISUALIZE - Understanding the geometric nature of the optimization problem
Let's solve for the intersection of our two main constraints. We'll use a simple substitution method:
From the second equation: \(\mathrm{4x + 5y = 28,000}\)
We can solve for x: \(\mathrm{x = \frac{28,000 - 5y}{4} = 7,000 - 1.25y}\)
Substitute into the first equation:
\(\mathrm{7(7,000 - 1.25y) + 6y = 38,000}\)
\(\mathrm{49,000 - 8.75y + 6y = 38,000}\)
\(\mathrm{49,000 - 2.75y = 38,000}\)
\(\mathrm{-2.75y = -11,000}\)
\(\mathrm{y = 4,000}\)
Now find x: \(\mathrm{x = 7,000 - 1.25(4,000) = 7,000 - 5,000 = 2,000}\)
So our intersection point is \(\mathrm{(2,000, 4,000)}\)
Let's check our key corner points:
Process Skill: MANIPULATE - Solving the system of equations systematically
Let's double-check that our points actually satisfy both constraints:
For \(\mathrm{(2,000, 4,000)}\):
For \(\mathrm{(0, 5,600)}\):
For \(\mathrm{(7,000, 0)}\):
The point \(\mathrm{(7,000, 0)}\) violates the first constraint, so it's not feasible.
Between our feasible points, \(\mathrm{(2,000, 4,000)}\) gives us the maximum total of \(\mathrm{x + y = 6,000}\).
Process Skill: APPLY CONSTRAINTS - Ensuring our solution respects all given limitations
The maximum number of balls and boxes combined that can be produced is 6,000.
This corresponds to producing 2,000 balls and 4,000 boxes, which uses all available resources optimally.
Answer: B. 6,000
1. Misunderstanding the optimization objective: Students often get confused about what they're trying to maximize. They might think they need to maximize either x (balls) OR y (boxes) separately, rather than understanding that the goal is to maximize the TOTAL \(\mathrm{(x + y)}\). This leads them to try finding maximum values for each variable individually instead of treating this as a linear programming problem.
2. Incorrectly handling the inequality constraints: Many students treat the inequalities \(\mathrm{(\leq)}\) as strict equalities \(\mathrm{(=)}\) from the start, not realizing that the maximum will occur when at least one constraint is "binding" (satisfied as an equality). They might solve each constraint separately as an equation without understanding that we need to find the intersection points of the feasible region.
3. Missing the geometric insight: Students often fail to recognize this as a linear programming problem where the optimal solution occurs at corner points of the feasible region. Instead, they might try algebraic manipulation without visualizing that they need to check boundary intersections and corner points systematically.
1. Algebraic errors when solving the system of equations: When finding the intersection point of \(\mathrm{7x + 6y = 38,000}\) and \(\mathrm{4x + 5y = 28,000}\), students frequently make mistakes in the substitution or elimination process. Common errors include sign errors (like getting \(\mathrm{+2.75y}\) instead of \(\mathrm{-2.75y}\)) or arithmetic mistakes when multiplying equations to eliminate variables.
2. Failing to check constraint feasibility: Students often forget to verify that their calculated corner points actually satisfy BOTH original constraints. For example, they might find that \(\mathrm{(7,000, 0)}\) gives \(\mathrm{x + y = 7,000}\) and assume this is the answer without checking that \(\mathrm{7(7,000) + 6(0) = 49,000 > 38,000}\), which violates the first constraint.
3. Incomplete corner point analysis: Many students only check one or two corner points instead of systematically evaluating all relevant intersections. They might find the intersection of the two main constraints but forget to check where each constraint intersects the axes \(\mathrm{(x = 0 \text{ and } y = 0 \text{ cases})}\).
1. Choosing an infeasible solution: After calculating that point \(\mathrm{(7,000, 0)}\) gives the highest sum of 7,000, students might select answer choice C (7,000) without realizing this point violates the first constraint. They focus only on which point gives the largest \(\mathrm{x + y}\) value without ensuring the point is actually achievable given the resource constraints.