Problem 14 - Applied Calculus

Question

A company's cost of producing \(q\) liters of a chemical is \(C(q)\) dollars; this quantity can be sold for \(R(q)\) dollars. Suppose \(C(2000)=5930\) and \(R(2000)=7780\) (a) What is the profit at a production level of \(2000 ?\) (b) If \(M C(2000)=2.1\) and \(M R(2000)=2.5,\) what is the approximate change in profit if \(q\) is increased from 2000 to \(2001 ?\) Should the company increase or decrease production from \(q=2000 ?\) (c) If \(M C(2000)=4.77\) and \(M R(2000)=4.32\) should the company increase or decrease production from \(q=2000 ?\)

Step-by-Step Solution

Answer

(a) $1850; (b) Increase production; (c) Decrease production.

Step 1: Calculate Profit at q=2000

To find the profit at a production level of 2000 liters, we need to subtract the total cost from the total revenue at this level. We use the formula: \(P(q) = R(q) - C(q)\). Plugging in the values provided, we get:\[P(2000) = R(2000) - C(2000) = 7780 - 5930 = 1850 \]The profit when 2000 liters are produced is $1850.

Step 2: Approximate Change in Profit for q=2001 using Marginal Costs and Revenues

We are given the Marginal Cost \(MC(2000) = 2.1\) and the Marginal Revenue \(MR(2000) = 2.5\). The change in profit when increasing production from 2000 to 2001 is the difference between \(MR\) and \(MC\).\[\Delta P = MR(2000) - MC(2000) = 2.5 - 2.1 = 0.4\]Thus, if production is increased by one unit, the profit will increase by approximately $0.40.

Step 3: Decision on Production Change at q=2000 with MC=2.1 and MR=2.5

Since \(MR(2000) = 2.5\) is greater than \(MC(2000) = 2.1\), the profit increases by $0.40 per additional unit produced. Therefore, the company should increase its production from 2000 liters.

Step 4: Decision on Production Change at q=2000 with MC=4.77 and MR=4.32

In this scenario, \(MC(2000) = 4.77\) and \(MR(2000) = 4.32\). Here, \(MC\) is greater than \(MR\), indicating that producing one more unit actually decreases profit.\[\Delta P = MR(2000) - MC(2000) = 4.32 - 4.77 = -0.45\]Thus, the company should decrease production from 2000 liters, as increasing it would decrease profit by $0.45 per unit.

Key Concepts

Profit Calculation

Profit calculation is essential for understanding how well a company is doing financially. It's a relatively straightforward process where we take the revenue, which is the income from selling goods, and subtract the total costs incurred in producing them. This is summarized by the formula:

• Profit, \(P(q)\) = Revenue, \(R(q)\) - Cost, \(C(q)\)

In our exercise, if the company produces 2000 liters, the total cost is \(5930\) dollars and the revenue is \(7780\) dollars. Therefore, the profit is calculated as \(7780 - 5930 = 1850\) dollars. This means that after selling 2000 liters, the company earns an additional \(1850\) dollars beyond covering its costs. Understanding profit helps businesses make strategic decisions, like evaluating market opportunities and optimizing resources.

Marginal Cost

Marginal Cost (MC) represents the cost of producing one additional unit of a product. It plays a critical role in decision-making as it informs businesses about the extra costs that will incur if production is increased. Knowing the MC allows a company to decide whether producing more is financially sensible or not.

• In the exercise, the marginal cost at the production level of 2000 liters is reported as \(2.1\) dollars, meaning that to produce one more liter, an extra \(2.1\) dollars would be spent.

It's important for businesses to keep the marginal costs low in comparison to the revenue gained from selling those additional units. This helps maintain a healthy profit margin while scaling production. The key is to identify when the cost of producing an extra unit is less than or equal to the revenue it generates.

Marginal Revenue

Marginal Revenue (MR) is the additional income received from selling one more unit of a product. It's crucial for analyzing how changes in the sales volume affect revenue and ultimately profit.

• For the company in the exercise, the marginal revenue at 2000 liters is \(2.5\) dollars per extra liter sold.

When the marginal revenue exceeds the marginal cost, each additional unit sold increases the company's overall profit. Understanding MR helps businesses determine the optimal level of production where profit is maximized. Businesses strive to operate where the marginal revenue equals or exceeds the marginal cost, ensuring maximum profitability with their available resources.

Production Decision-making

Production decision-making involves determining the optimal quantity of a product to produce in order to maximize profit. The process is intrinsically linked with both marginal cost and marginal revenue concepts.

• If MR is greater than MC, as in the first scenario of the exercise, the company should increase production because additional units are profitable. Specifically, when MC is \(2.1\) and MR is \(2.5\), each unit adds \(0.4\) dollars to the profit.
• On the contrary, if MC exceeds MR, like in the second scenario with MC as \(4.77\) and MR as \(4.32\), the company should decrease production. In this case, additional units would reduce the profit by \(-0.45\) dollars each.

Making effective production decisions involves analyzing these marginal figures to align production levels with profit goals. This ensures that resources are utilized efficiently and that the business is operating near its optimal capacity.