Math Problem Statement

A company's average cost per unit when x units are produced is defined to be

Average cost =

Total cost

x

.

Suppose a company's average costs are given by

Average cost =

5000/x + 75 + 0.8x.

Solution

The company's average cost per unit formula is given by:

Average Cost=Total Costx\text{Average Cost} = \frac{\text{Total Cost}}{x}

where xx is the number of units produced.

In this case, the average cost function is already provided as:

Average Cost=5000x+75+0.8x\text{Average Cost} = \frac{5000}{x} + 75 + 0.8x

This function gives the average cost of producing xx units, where:

  • 5000x\frac{5000}{x} represents the fixed costs distributed over xx units,
  • 7575 is a constant cost per unit,
  • 0.8x0.8x is the variable cost that increases with each additional unit produced.

Further Questions:

  1. What is the total cost function based on the given average cost function?
  2. How does the average cost change as xx increases?
  3. What is the minimum average cost and at what production level does it occur?
  4. How would the average cost function change if the fixed cost was increased to 60006000?
  5. How does the variable cost 0.8x0.8x affect the overall cost as production scales?

Tip: The total cost function can be derived by multiplying the average cost by xx, giving insight into the cost behavior as production changes.

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Math Problem Analysis

Mathematical Concepts

Algebra
Cost Functions
Optimization

Formulas

Average Cost = Total Cost / x
Average Cost = 5000/x + 75 + 0.8x

Theorems

Optimization of Functions
Cost Function Analysis

Suitable Grade Level

Grades 11-12