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) To write the cost function, we need to consider both the fixed cost and the marginal cost per pair of shoes. The fixed cost is $54,000, which does not change regardless of the number of shoes produced. The marginal cost is $25 per pair. Therefore, the cost function can be expressed as:
Cost function = Fixed cost + (Marginal cost * Number of pairs)
Cost function = $54,000 + ($25 * Number of pairs)
b) To write the revenue function, we need to consider the selling price per pair and the number of pairs sold. The selling price per pair is $115. Therefore, the revenue function can be expressed as:
Revenue function = Selling price per pair * Number of pairs
Revenue function = $115 * Number of pairs
c) To find the number of shoes that must be sold for the company to break even, we need to equate the cost function to the revenue function. In other words, when the total revenue equals the total cost, the company breaks even. So, we set:
Revenue function = Cost function
$115 * Number of pairs = $54,000 + ($25 * Number of pairs)
Simplifying the equation, we get:
$115 * Number of pairs - $25 * Number of pairs = $54,000
$90 * Number of pairs = $54,000
Number of pairs = $54,000 / $90
Number of pairs = 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 equal to the difference between the revenue and cost:
Profit = Revenue function - Cost function
$180,000 = ($115 * Number of pairs) - ($54,000 + ($25 * Number of pairs))
Simplifying the equation, we get:
$180,000 + $54,000 = $115 * Number of pairs - $25 * Number of pairs
$234,000 = $90 * Number of pairs
Number of pairs = $234,000 / $90
Therefore, the company must sell approximately 2,600 pairs of shoes to make a profit of $180,000.