Application of function in business and economics

Student Apozee

Student

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A firm sells a single product for $65 per unit. Variable costs per unit are $20 for materials and $27 for labor. Annual fixed cost is $100,000. Construct the profit function stated in terms of x, the number of units produced and sold. What profit is earned if annual sales are 15,000 units?

  • 24 Dec, 2023
Replies (2)
Student Apozee

Student

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The profit function can be expressed as the difference between total revenue and total cost. Total revenue is given by the product of the selling price and the number of units sold, and total cost is the sum of fixed costs and variable costs.

Let's denote the following:

  • �(�): Total revenue
  • �(�): Total cost
  • �(�): Profit

Given:

  • Selling price per unit (�): $65
  • Variable cost for materials per unit (��V1): $20
  • Variable cost for labor per unit (��): $27
  • Annual fixed cost (�): $100,000
  • Number of units produced and sold (�): The variable we want to express the profit in terms of.

Now, let's construct the profit function:

R(x) = S.x

C(x) = F+V1x+V2x

�(�)=�⋅�


P(X)=R(x) - C(x)

Substitute the given values into the equations:

�(�)=65⋅�R(x) = 65x

C(x) = 100000 + 20x+27x

Now, substitute �(�) and �(�) into the profit function:

�(�)=65⋅�−(20⋅�+27⋅�+100,000)P(X)=65x - (100000 + 20x+27x)

Simplify the expression:

�(�)=65�−47�−100,000P(X)=18x - 100000

Now, we have the profit function, P(X)=18x - 100000

�(�) in terms of the number of units produced and sold (�).

To find the profit earned if annual sales are 15,000 units, substitute x = 15 into the profit function:

�(15,000)=18⋅15,000−100,000P(15000)=18. 15000 - 100000 = $170000

Therefore, the profit earned if annual sales are 15,000 units is $170,000.

  • 24 Dec, 2023

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