Problem 13 - 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) Profit is $1850. (b) Increase production; profit increases by $0.4. (c) Decrease production; costs exceed revenue.

Step 1: Calculate Profit at Production Level 2000

Profit is calculated as the difference between revenue and cost. Given: - Cost of producing 2000 liters: \(C(2000) = 5930\)- Revenue from selling 2000 liters: \(R(2000) = 7780\)The profit at production level 2000 is:\[ P(2000) = R(2000) - C(2000) = 7780 - 5930 = 1850 \]

Step 2: Calculate Approximate Change in Profit for Increase in Production to 2001

The approximate change in profit is computed using the difference between marginal revenue and marginal cost, multiplied by the change in quantity. Given:- Marginal Cost at 2000 liters: \(MC(2000) = 2.1\)- Marginal Revenue at 2000 liters: \(MR(2000) = 2.5\)- Change in \(q\): 1 literThe change in profit is:\[ \Delta P \approx (MR(2000) - MC(2000)) \times \Delta q = (2.5 - 2.1) \times 1 = 0.4 \]

Step 3: Determine if Company Should Increase or Decrease Production (MC=2.1, MR=2.5)

Since the marginal revenue \(MR(2000) = 2.5\) is greater than the marginal cost \(MC(2000) = 2.1\), this means that increasing production from 2000 will increase profit. Therefore, the company should increase production.

Step 4: Determine if Company Should Increase or Decrease Production (MC=4.77, MR=4.32)

Given:- New Marginal Cost at 2000 liters is \(MC(2000) = 4.77\)- New Marginal Revenue at 2000 liters is \(MR(2000) = 4.32\)Since the marginal cost is greater than the marginal revenue (\(4.77 > 4.32\)), increasing production would decrease profit. Therefore, the company should decrease production.

Key Concepts

Profit Calculation

Calculating profit is a fundamental aspect of any business, as it represents the financial gain that the company makes. Profit is simply the difference between revenue, which is the money made from selling goods, and cost, which encompasses all expenses incurred in producing those goods.
In the example exercise, the company produces 2000 liters of a chemical, resulting in a production cost of \(C(2000) = 5930\) dollars, and sells it for a revenue of \(R(2000) = 7780\) dollars. Therefore, the profit at this production level is calculated using the formula:
\[ P(2000) = R(2000) - C(2000) = 7780 - 5930 = 1850 \]
This equation shows that the company achieves a profit of 1850 dollars by producing and selling 2000 liters of the chemical. Understanding this basic calculation can guide business decisions by indicating whether production levels should be maintained, increased, or decreased.

Marginal Cost

Marginal cost is a crucial concept in economics and business that represents the cost of producing one additional unit of a good. Calculating the marginal cost helps companies make informed decisions because it reflects the additional expenses that occur with increased production.
In our case, when considering the production of the chemical at 2000 liters, the marginal cost is given as \(MC(2000) = 2.1\) dollars. This means it costs the company an additional 2.1 dollars to produce each extra liter beyond the 2000th liter.

• It helps businesses decide if expanding production is financially wise.
• By comparing marginal cost to marginal revenue, companies can optimize their profit.

If the marginal cost starts exceeding the marginal revenue, future production may become less profitable, prompting a re-evaluation of production strategies.

Marginal Revenue

Marginal revenue refers to the additional income generated from selling one more unit of a good or service. It is a vital measure that helps companies understand how much extra income they gain as they increase their sales volume.
For the chemical production at 2000 liters, the marginal revenue is \(MR(2000) = 2.5\) dollars. This indicates that by selling one additional liter, the company earns 2.5 dollars more.
Understanding marginal revenue assists businesses in evaluating whether increasing sales will be beneficial. When marginal revenue is greater than marginal cost, as in the initial scenario where \(MR(2000) = 2.5\) dollars and \(MC(2000) = 2.1\) dollars:

• Extra production increases profit.
• Businesses are encouraged to boost production.

However, if the marginal cost is higher, profits may diminish, as seen when \(MC(2000) = 4.77\) is higher than \(MR(2000) = 4.32\). Understanding these dynamics helps companies adjust their production levels strategically for optimal profitability.