Question

Make a real world function that models the relationship between the two quantities. The function should be a piecewise function where not all components are linear. Describe the scenario you are modeling, the definitions of the variables involved in the function, and a graph or equation of the function.

Answer 1

Scenario

Let's model the relationship between the total cost of producing handcrafted furniture and the number of pieces produced. In our scenario, a small workshop handcrafts wooden furniture such as chairs and tables. The cost structure is affected by the number of items produced due to economies of scale for larger productions and fixed costs for smaller productions.

Variables

• x: Number of furniture pieces produced
• C(x): Total cost of producing x pieces of furniture, in dollars

Piecewise Function Definition

The cost function \( C(x) \) can be modeled as a piecewise function:

\[ C(x) = \begin{cases} 100 + 20x & \text{if } 0 \leq x \leq 10 \ 100 + 15x & \text{if } 11 \leq x \leq 50 \ 100 + 10x & \text{if } x > 50 \end{cases} \]

Explanation of the Pieces

1. For \( 0 \leq x \leq 10 \):

   • Cost Function: \( C(x) = 100 + 20x \)
   • Interpretation: In this range, there are high fixed costs and the cost increases by $20 for each additional piece produced due to lower productivity and more customization.

2. For \( 11 \leq x \leq 50 \):

   • Cost Function: \( C(x) = 100 + 15x \)
   • Interpretation: In this range, the workshop begins to achieve some economies of scale, reducing the cost per item to $15 as production increases.

3. For \( x > 50 \):

   • Cost Function: \( C(x) = 100 + 10x \)
   • Interpretation: At this level of production, the workshop benefits significantly from economies of scale, reducing the cost to produce each additional piece to $10.

Graph of the Function

While I cannot create a visual graph directly here, I can describe what it looks like:

• On a graph where the x-axis represents the number of furniture pieces (from 0 to over 100) and the y-axis represents the cost (in dollars), the function consists of three linear segments:
  • The first segment slopes with a steep incline, starting at $(0, 100)$ and ending at approximately $(10, 300)$.
  • The second segment, starting from $(11, 285)$ to $(50, 800)$, is less steep.
  • The final segment from $(51, 610)$ continues to rise, but at a gentler slope compared to the previous segments.

Conclusion

This piecewise function effectively captures the real-world scenario of a handcrafted furniture workshop, illustrating how varying production levels affect costs due to fixed costs, productivity variations, and economies of scale. Each segment of the piecewise function reflects different production strategies and cost implications.