Optimising Production Cost Function

00:00 7 marks

A manufacturing company wishes to minimise production costs for a new product. The cost is thought to follow a quadratic trend due to fixed costs, variable costs, and an increasing marginal cost as production scales. The following multi-part question addresses setting up a polynomial cost function and determining the optimal production level.

Proceed with the parts below:

Part a

6 minutes 4 marks

Question

A local manufacturer is planning to produce a new line of custom phone cases. The cost of production is influenced by fixed costs, variable costs per unit, and additional costs due to scaling production (such as overtime or increased waste). Propose a polynomial function of the form \(C(x) = ax^2 + bx + c\) that might represent the total cost (in dollars) for producing \(x\) cases. In your response, clearly explain the role of each coefficient and constant in the function within the context of production costs.

Hint Hint

Hint

Consider that the constant term \(c\) represents fixed costs, \(b\) represents variable cost per unit, and \(a\) indicates increasing marginal costs due to scaling.

Answer Answer
Mark Scheme Mark Scheme
Explanation Explanation
Formula Booklet

Answer Answer

Complete Answer

One possible model is \(C(x) = 0.05x^2 + 20x + 500\), where \(500\) represents fixed costs, \(20\) represents the cost per unit produced, and \(0.05\) captures the effect of increasing marginal costs as production increases.

Mark Scheme Mark Scheme

Answer Explanation Answer Explanation

Formula Booklet

Part b

5 minutes 3 marks

Question

Using your cost function from part (a), determine the level of production \(x\) that minimizes the cost. Do this by finding the derivative \(C'(x)\), setting it equal to zero, and solving for \(x\).

Hint Hint

Hint

Differentiate the function \(C(x) = ax^2 + bx + c\) to obtain \(C'(x)\) and solve \(C'(x) = 0\). Remember to consider the practical domain for production quantity (\(x \geq 0\)).

Answer Answer
Mark Scheme Mark Scheme
Explanation Explanation
Formula Booklet

Answer Answer

Complete Answer

For the function \(C(x) = 0.05x^2 + 20x + 500\), the derivative is \(C'(x) = 0.1x + 20\). Setting the derivative equal to zero yields \(0.1x + 20 = 0\), so \(x = -200\). Since production quantity cannot be negative, the minimum cost within the feasible domain \(x \geq 0\) occurs at \(x=0\).

Mark Scheme Mark Scheme

Answer Explanation Answer Explanation

Formula Booklet