Problem 46 - Precalculus: Mathematics for Calculus

Question

Kitchen Korner produces refrigerators, dishwashers, and stoves at three different factories. The table gives the number of each product produced at each factory per day. Kitchen Korner receives an order for 110 refrigerators, 150 dishwashers, and 114 ovens. How many days should each plant be scheduled to fill this order? $$\begin{array}{|l|c|c|c|}\hline \text { Appliance } & \text { Factory A } & \text { Factory B } & \text { Factory C } \\\\\hline \text { Refrigerators } & 8 & 10 & 14 \\\\\text { Dishwashers } & 16 & 12 & 10 \\\\\text { Stoves } & 10 & 18 & 6 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

Factory A: 4 days, Factory B: 6 days, Factory C: 2 days.

Step 1: Understand the Problem

The task is to find how many days each factory should work to meet the production demand. We know the daily production rates from the table, and we must fulfill the order by finding how long each factory needs to operate to meet or exceed the order requirements for refrigerators, dishwashers, and stoves.

Step 2: Set Up Variables

Let's denote the number of days Factory A, B, and C needs to operate as \( x \), \( y \), and \( z \) respectively. Our goal is to find the values of \( x \), \( y \), and \( z \) to meet the appliance order.

Step 3: Formulate the Equations

Based on the daily production rates, the equations are derived from the order requirements: \( 8x + 10y + 14z = 110 \) for refrigerators, \( 16x + 12y + 10z = 150 \) for dishwashers, and \( 10x + 18y + 6z = 114 \) for stoves.

Step 4: Solve the System of Equations

To find the values of \( x \), \( y \), and \( z \), solve the system of equations: \( 8x + 10y + 14z = 110 \), \( 16x + 12y + 10z = 150 \), and \( 10x + 18y + 6z = 114 \). Use substitution or elimination methods to solve these equations.

Step 5: Calculate Specific Values

After solving, you find: \( x = 4 \), \( y = 6 \), and \( z = 2 \). These values indicate the days each factory should be scheduled to fulfill the order.

Key Concepts

Production Scheduling

Production scheduling is a crucial aspect of managing manufacturing expectations, especially when orders involve multiple products across different production lines. In the case of Kitchen Korner, we need to schedule the production in such a way that the total order demand is met with minimal delay and cost.

• Each factory has a specific daily production rate for refrigerators, dishwashers, and stoves.
• We aim to determine the number of days each factory should operate to meet the production demands for each appliance.
• The goal is to align the production capacity of each factory with the order requirements, ensuring efficient use of resources.

Understanding production scheduling helps optimize the workflow by balancing output with demand, thus avoiding overproduction or resource wastage. By carefully planning the number of working days for each factory, Kitchen Korner can efficiently complete the orders.

Linear Algebra

Linear algebra plays a pivotal role in solving systems of equations, which are common in production and scheduling problems. In Kitchen Korner's scenario, it involves converting the task into a mathematical model using a system of linear equations.

• The equations are derived from the daily production rates and the total number required for each appliance.
• For instance, the equation for refrigerators, \( 8x + 10y + 14z = 110 \), reflects the contribution of each factory's production to the total demand.
• The other products, dishwashers and stoves, follow similarly structured equations.

By structuring the problem this way, we utilize linear algebra to systematically discover the optimum number of days each factory should work. We employ techniques like the substitution method or elimination method to solve the set of equations, providing a clear path to the desired solution.

Problem-Solving in Precalculus

Problem-solving in precalculus often involves translating real-world scenarios into mathematical problems, as seen in Kitchen Korner's scheduling issue. The ability to interpret data correctly and form equations is a skill honed in precalculus.

• Identify the problem - the need to fulfill specific orders using various production capacities.
• Formulate the problem - converting production capabilities into linear equations reflecting real-world constraints and objectives.
• Solve the problem - applying methods learned in precalculus, such as solving equations, interpreting solutions, and verifying results.

Approaching problems systematically using these precalculus techniques enables us to not only solve the equations effectively but also understand the interconnectedness of variables in practical scenarios like production and scheduling. It's about maximizing efficiency while meeting specified goals.