Math Problem Statement

The processing and shipping cost C (in thousands of dollars) for ordering the components used in manufacturing a product is given by C = 100

200 x2 + x x + 30

, x ≥ 1 where x is the order size (in hundreds of components). Use a graphing utility to graph the cost function.

From the graph, estimate the order size that minimizes cost. (Round your answer to the nearest whole number.) . components

Solution

The graph above represents the cost function C(x)C(x). By examining it visually, we can estimate the order size xx that minimizes the cost.

From the graph, it appears that the minimum cost is reached at an order size close to x=20x = 20 (in hundreds of components). Therefore, the optimal order size that minimizes cost is approximately:

x20×100=2000 componentsx \approx 20 \times 100 = 2000 \text{ components}

Would you like more details on finding the exact minimum value, or do you have further questions?


Here are some additional questions based on this problem:

  1. How would changing the base cost C(100)C(100) impact the order size for the minimum cost?
  2. What is the impact on cost if the order size xx is doubled from the minimum order size?
  3. How does the term xx+30\frac{x}{x + 30} affect the shape of the cost function for large values of xx?
  4. Can the minimum cost order size change if the condition x1x \geq 1 is modified?
  5. How would using a different model for cost estimation affect the optimal order size?

Tip: When analyzing cost functions, always consider the impact of each term separately to understand their effects on the overall cost.

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Math Problem Analysis

Mathematical Concepts

Algebra
Calculus
Optimization

Formulas

Cost function C(x) = 100 - (200/x^2) + (x/(x + 30))

Theorems

Optimization through calculus (finding minima)

Suitable Grade Level

Grades 10-12