. A cement manufacturer has a cost of production of $28 for each bag of cement produced and the sells a bag of cement for $40. The monthly fixed cost of the manufacturer is $180,000.
a. Derive the monthly cost function of the manufacturer?
b. Derive the monthly revenue function of the manufacturer?
c. Derive the monthly profit function of the manufacturer?
d. How many bags must be produced and sold to make a profit of $12,000?
e. Compute the profit or loss which corresponds to production levels of 14000 bags of cement.
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a. To derive the monthly cost function, we need to consider both the variable cost and the fixed cost. The variable cost is the cost of producing each bag of cement, which is given as $28.
So, the monthly cost function can be expressed as:
Cost = Fixed Cost + (Variable Cost per unit * Number of units)
Given that the monthly fixed cost is $180,000 and the variable cost per unit is $28, the monthly cost function can be written as:
Cost = 180,000 + (28 * Number of units)
b. The monthly revenue function can be derived by multiplying the selling price per bag with the number of bags sold. The selling price per bag is $40.
So, the monthly revenue function can be written as:
Revenue = Selling Price per unit * Number of units
Given that the selling price per bag is $40, the monthly revenue function can be expressed as:
Revenue = 40 * Number of units
c. The monthly profit function can be derived by subtracting the monthly cost function from the monthly revenue function. So,
Profit = Revenue - Cost
Profit = (40 * Number of units) - (180,000 + (28 * Number of units))
Profit = 40 * Number of units - 28 * Number of units - 180,000
d. To find the number of bags that must be produced and sold to make a profit of $12,000, we can set the profit function equal to $12,000 and solve for the number of units.
40 * Number of units - 28 * Number of units - 180,000 = 12,000
12 * Number of units = 192,000
Number of units = 192,000 / 12
Number of units = 16,000 bags
Therefore, to make a profit of $12,000, 16,000 bags must be produced and sold.
e. To compute the profit or loss at a production level of 14,000 bags, we can substitute the value of 14,000 into the profit function.
Profit = 40 * Number of units - 28 * Number of units - 180,000
Profit = 40 * 14,000 - 28 * 14,000 - 180,000
Profit = 560,000 - 392,000 - 180,000
Profit = -12,000
At a production level of 14,000 bags, the manufacturer would incur a loss of $12,000.