A manufacturer sells two products, one at a price of $3000 a unit and the other at a price of $12000 a unit. A quantity q1 of the first product and q2 of the second product are sold at a total cost of $5000 to the manufacturer.
Express the manufacturer's profit, as a function of q1 and q2.
To express the manufacturer's profit as a function of q1 and q2, we need to determine the cost of production for each product and subtract it from the total revenue.
Let's denote the cost of production for the first product as c1 and the cost of production for the second product as c2.
The revenue from selling the first product is given by q1 multiplied by the price per unit ($3000), so the revenue for the first product is 3000q1.
Similarly, the revenue from selling the second product is given by q2 multiplied by the price per unit ($12000), so the revenue for the second product is 12000q2.
The total revenue from both products is the sum of the revenues from each product, so the total revenue is 3000q1 + 12000q2.
Now, the cost of production for the first product is given as $3000 per unit, so the cost of production for q1 units of the first product is 3000q1.
Similarly, the cost of production for the second product is given as $12000 per unit, so the cost of production for q2 units of the second product is 12000q2.
The total cost of production for both products is the sum of the costs of production for each product, so the total cost of production is 3000q1 + 12000q2.
Finally, the profit is calculated by subtracting the total cost of production from the total revenue, so the profit function is:
Profit(q1, q2) = (3000q1 + 12000q2) - (3000q1 + 12000q2) = 0.
Therefore, the manufacturer's profit depends on the number of units sold but remains constant at $0 regardless of the quantity of each product sold.
To express the manufacturer's profit as a function of q1 and q2, we need to calculate the revenue and cost associated with each product.
Revenue for the first product (q1) can be calculated by multiplying the price ($3000) by the quantity (q1):
Revenue from first product = 3000 * q1
Revenue for the second product (q2) can be calculated similarly:
Revenue from second product = 12000 * q2
Now, let's calculate the total revenue:
Total Revenue = Revenue from first product + Revenue from second product
= 3000 * q1 + 12000 * q2
To determine the cost, we need to consider that the total cost to the manufacturer is $5000. We can assume a portion of this cost is associated with each product based on their respective quantities:
Let's assume the cost for the first product (C1) is proportional to the quantity (q1), and the cost for the second product (C2) is proportional to the quantity (q2). The cost for each product can be calculated using the following formulas:
C1 = (q1 / (q1 + q2)) * 5000
C2 = (q2 / (q1 + q2)) * 5000
The total cost to the manufacturer is the sum of these costs:
Total Cost = C1 + C2
= (q1 / (q1 + q2)) * 5000 + (q2 / (q1 + q2)) * 5000
Finally, we can calculate the profit by subtracting the total cost from the total revenue:
Profit = Total Revenue - Total Cost
= 3000 * q1 + 12000 * q2 - ((q1 / (q1 + q2)) * 5000 + (q2 / (q1 + q2)) * 5000)
Hence, the manufacturer's profit is given by the function:
Profit(q1, q2) = 3000 * q1 + 12000 * q2 - ((q1 / (q1 + q2)) * 5000 + (q2 / (q1 + q2)) * 5000)