Problem 27 - Intermediate Algebra

Question

Production Planning. \(\quad\) A manufacturer builds racing bikes and mountain bikes, with the per-unit manufacturing costs shown in the table. The company has budgeted \(\$ 26,150\) for materials and \(\$ 31,800\) for labor. How many bicycles of each type can be built? $$ \begin{array}{|l|c|c|} \hline \text { Model } & \text { Cost of materials } & \text { Cost of labor } \\\ \hline \text { Racing } & \$ 110 & \$ 120 \\ \text { Mountain } & \$ 140 & \$ 180 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The manufacturer can build 85 racing bikes and 120 mountain bikes.

Step 1: Define Variables

Let \( x \) be the number of racing bikes and \( y \) be the number of mountain bikes. Our goal is to find how many of each type can be manufactured given the budget constraints.

Step 2: Set Up Equations for Materials

The cost equation for materials is \( 110x + 140y = 26150 \). This represents the budget limit for materials.

Step 3: Set Up Equations for Labor

The cost equation for labor is \( 120x + 180y = 31800 \). This equation represents the budget limit for labor.

Step 4: Solve the System of Equations

We use the method of substitution or elimination to solve the system of equations: 1. \( 110x + 140y = 26150 \) 2. \( 120x + 180y = 31800 \). Simplifying 1 gives: \( 11x + 14y = 2615 \) and simplifying 2 gives \( 12x + 18y = 3180 \). Multiply the first equation by 6 and the second by 5: \[ 66x + 84y = 15690 \] \[ 60x + 90y = 15900 \]. Solve using elimination.

Step 5: Eliminate and Solve for One Variable

Subtract the equations: \( (66x + 84y) - (60x + 90y) = 15690 - 15900 \) which simplifies to: \( 6x - 6y = -210 \). Divide by 6 to get: \( x - y = -35 \).

Step 6: Substitute to Find Values

From \( x - y = -35 \), express \( x \) as \( x = y - 35 \). Substitute \( x = y - 35 \) into \( 11x + 14y = 2615 \):\( 11(y-35) + 14y = 2615 \).Simplify to get: \( 11y - 385 + 14y = 2615 \), which further simplifies to \( 25y = 3000 \). Solving gives \( y = 120 \).

Step 7: Solve for the Other Variable

Substitute \( y = 120 \) back into \( x = y - 35 \) to find \( x \). Thus, \( x = 120 - 35 = 85 \).

Step 8: Verify the Solution

Recalculate both equations with \( x = 85 \) and \( y = 120 \): 1. \( 110(85) + 140(120) = 26150 \) - correct. 2. \( 120(85) + 180(120) = 31800 \) - correct. Both budget constraints are met.

Key Concepts

Production Planning

Production planning is a key process in manufacturing that involves determining how much of each product should be produced to meet customer demand and business constraints. In our exercise, we are planning the production of racing and mountain bikes. The manufacturer needs to consider several factors:

• The cost of materials for each bike model
• The cost of labor to produce each bike model
• The total budget available for materials and labor

Good production planning ensures that resources are used effectively and that products are manufactured without exceeding budget limits.
This also involves creating a strategy that considers both product costs and available production capabilities.

Systems of Equations

Systems of equations are a set of two or more equations with the same variables. In our context, these equations represent the relationships between the number of bikes produced and the cost constraints for materials and labor.
In the example, our system of equations starts with:

• Material costs: \(110x + 140y = 26150\)
• Labor costs: \(120x + 180y = 31800\)

Here, \(x\) is the number of racing bikes and \(y\) is the number of mountain bikes. To solve a system of equations, methods such as substitution or elimination are often used. Solving these equations allows us to find the exact number of each type of bike that can be produced without exceeding the given budgets. By aligning production with cost constraints, manufacturers can optimize resource use.

Budget Constraints

Budget constraints are limits on the spending capacity of a business. They define how much can be spent in particular areas, such as materials and labor in our exercise. Meeting budget constraints is crucial to sustainable business operations.
In the bike production exercise, we tackled the following budget constraints:

• Materials cost cannot exceed \(\\(26,150\)
• Labor cost cannot exceed \(\\)31,800\)

These constraints lead us to set up specific equations that define the limits on how many bikes can be produced. Budget constraints require manufacturers to make strategic decisions in production planning to ensure that costs do not outstrip the finances available. Constraints like these can force businesses to find efficient ways of production, balancing costs while meeting market demand.