A shoe company will make a new type of running shoe. The fixed cost for the production will be $54,000. The marginal cost will be $25 per pair of shoes. The shoes will sell for $115 for each pair.
a) Write the cost function
b) Write the revenue function
c) Find the number of shoes that must be sold for the company to break even.
d) How many shoes must be sold for the company to profit $180,000.
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.
a) The cost function is the sum of the fixed cost and the product of the marginal cost and the number of shoes produced. In this case, the fixed cost is $54,000, and the marginal cost is $25 per pair of shoes.
The cost function can be written as:
Cost = $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. In this case, the selling price per pair of shoes is $115.
The revenue function can be written as:
Revenue = $115 * Number of shoes
c) To find the number of shoes that must be sold for the company to break even, we need to find the point where the revenue equals the cost.
Revenue = Cost
$115 * Number of shoes = $54,000 + ($25 * Number of shoes)
Simplifying the equation:
$115 * Number of shoes - $25 * Number of shoes = $54,000
Combining like terms:
$90 * Number of shoes = $54,000
Dividing both sides by $90:
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 can modify the equation to represent the profit.
Profit = Revenue - Cost
$180,000 = $115 * Number of shoes - ($54,000 + ($25 * Number of shoes))
Simplifying the equation:
$180,000 = $115 * Number of shoes - $54,000 - $25 * Number of shoes
Combining like terms:
$180,000 = $90 * Number of shoes - $54,000
Adding $54,000 to both sides:
$180,000 + $54,000 = $90 * Number of shoes
$234,000 = $90 * Number of shoes
Dividing both sides by $90:
Number of shoes = $234,000 / $90
Number of shoes ≈ 2,600
Therefore, the company must sell approximately 2,600 pairs of shoes to profit $180,000.