Problem 6 - Applied Calculus

Question

For \(q\) units of a product, a manufacturer's cost is \(C(q)\) dollars and revenue is \(R(q)\) dollars, with \(C(500)=\) \(7200, R(500)=9400, M C(500)=15\), and \(M R(500)=20\) (a) What is the profit or loss at \(q=500 ?\) (b) If production is increased from 500 to 501 units, by approximately how much does profit change?

Step-by-Step Solution

Answer

(a) Profit is $2200 at q=500. (b) Profit increases by $5 when production increases from 500 to 501 units.

Step 1: Define Profit Formula

The profit is the revenue minus the cost, denoted by the formula:\[ P(q) = R(q) - C(q) \]We need to use this formula to find the profit at \(q = 500\).

Step 2: Calculate Profit at q=500

Substitute the values from the problem into the profit formula:\[ P(500) = R(500) - C(500) = 9400 - 7200 = 2200 \]This means the profit at \(q = 500\) is \(2200\) dollars.

Step 3: Define Marginal Profit

Marginal profit is given by the difference between marginal revenue and marginal cost. It represents the change in profit when one additional unit is produced:\[ MP(q) = MR(q) - MC(q) \]

Step 4: Calculate Change in Profit from 500 to 501 Units

Substitute the given values into the marginal profit equation:\[ MP(500) = MR(500) - MC(500) = 20 - 15 = 5 \]This means that if production is increased from 500 to 501 units, the profit increases by approximately 5 dollars.

Key Concepts

Profit Calculation

Profit is one of the key measures to determine the financial success of any production endeavor. It assesses how much money is made after accounting for the costs incurred in production. The formula for calculating profit is quite simple: subtract the total cost from the total revenue. Mathematically, it is expressed as:

• \[ P(q) = R(q) - C(q) \]

where \( P(q) \) is the profit, \( R(q) \) is the revenue, and \( C(q) \) is the cost for producing \( q \) units.
The exercise provided values to find the profit when 500 units are produced:

• \( R(500) = 9400 \)
• \( C(500) = 7200 \)

Substituting these values into the formula gives us:

• \[ P(500) = 9400 - 7200 = 2200 \]

Thus, at 500 units, the profit is 2200 dollars. This calculation helps understand how profitable a business can be at a certain production level.

Marginal Cost

Marginal cost is a critical concept in economics. It measures the cost of producing one additional unit of a product. Understanding marginal cost helps businesses make decisions about increasing or reducing production quantities.
In mathematical terms, marginal cost, represented as \( MC(q) \), can be seen as the rate of change of the total cost function as production increases by an incremental unit. In this exercise, the marginal cost at 500 units is:

• \( MC(500) = 15 \)

This indicates that producing the 501st unit will add 15 dollars to the total cost. Businesses often use marginal cost to determine the optimal level of production that minimizes costs and maximizes profits. Comparing marginal cost to marginal revenue helps in making informed production decisions.

Marginal Revenue

Marginal revenue is an important measure in determining how much additional revenue is generated from selling one more unit of a product. It helps businesses understand how revenue will change with different levels of production.
Mathematically, marginal revenue \( MR(q) \) is the derivative of the revenue function with respect to quantity. In this specific exercise, the marginal revenue at 500 units is given as:

• \( MR(500) = 20 \)

This value suggests that selling the 501st unit would bring in an additional 20 dollars in revenue.
Using marginal revenue, along with marginal cost, businesses can perform a marginal analysis. This helps determine the effect on profit when deciding to increase production by one unit. In this case, the marginal profit can be calculated as the difference between marginal revenue and marginal cost:

• \[ MP(500) = MR(500) - MC(500) = 20 - 15 = 5 \]

This shows that increasing production from 500 to 501 units would increase the profit by roughly 5 dollars, providing useful insights into production decisions.