a) The cost function can be calculated by adding the fixed cost to the variable cost. The variable cost is the product of the marginal cost per pair of shoes and the number of shoes produced.
Cost function = Fixed cost + (Marginal cost per pair of shoes × Number of shoes)
Cost function = $54,000 + ($25 × Number of shoes)
b) The revenue function is the product of the selling price per pair of shoes and the number of shoes sold.
Revenue function = Selling price per pair of shoes × Number of shoes
Revenue function = $115 × Number of shoes
c) To find the number of shoes that must be sold for the company to break even, we need to set the cost function equal to the revenue function.
Cost function = Revenue function
$54,000 + ($25 × Number of shoes) = $115 × Number of shoes
Simplifying the equation:
$54,000 + $25 × Number of shoes = $115 × Number of shoes
$54,000 = $115 × Number of shoes - $25 × Number of shoes
$54,000 = $90 × Number of shoes
Number of shoes = $54,000 / $90
Number of shoes = 600
Therefore, the company must sell 600 pairs of shoes to break even.
d) To find the number of shoes that must be sold for the company to profit $180,000, we need to set the profit (revenue - cost) equal to $180,000.
Profit = Revenue - Cost
$180,000 = ($115 × Number of shoes) - ($54,000 + ($25 × Number of shoes))
$180,000 = $115 × Number of shoes - $54,000 - $25 × Number of shoes
$180,000 = $90 × Number of shoes - $54,000
Simplifying the equation:
$90 × Number of shoes = $180,000 + $54,000
$90 × Number of shoes = $234,000
Number of shoes = $234,000 / $90
Number of shoes ≈ 2,600
Therefore, the company must sell approximately 2,600 pairs of shoes to make a profit of $180,000.