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 \(C(500)=\) \(7200, \quad R(500)=9400, \quad M C(500) \quad=\quad 15, \quad\) 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 at 500 units is 2200 dollars. (b) Profit increases by about 5 dollars for 501 units.

Step 1: Calculate Profit at q=500

To determine the profit at 500 units, use the formula for profit, which is the difference between revenue and cost: \( P(q) = R(q) - C(q) \). From the information given: \( R(500) = 9400 \) and \( C(500) = 7200 \). Substitute these values into the formula to find \( P(500) \): \[ P(500) = 9400 - 7200 = 2200 \] Thus, the profit at 500 units is 2200 dollars.

Step 2: Determine the Change in Revenue and Cost

When production increases from 500 to 501 units, the marginal values indicate the change in revenue and cost. The marginal cost \( MC(500) \) is 15, implying the cost increases by 15 dollars for an additional unit. Similarly, the marginal revenue \( MR(500) \) is 20, implying the revenue increases by 20 dollars for an additional unit.

Step 3: Calculate Change in Profit from q=500 to q=501

The change in profit \( \Delta P \) when increasing production from 500 to 501 units is given by the change in revenue minus the change in cost: \[ \Delta P = \Delta R - \Delta C \] Substitute the given marginal values: \[ \Delta P = 20 - 15 = 5 \] Thus, the profit increases by approximately 5 dollars when production is increased from 500 to 501 units.

Key Concepts

Understanding Marginal Cost

Marginal cost is a key concept in economics that describes the change in total cost when producing one additional unit of a good or service. It's useful for manufacturers to understand how their costs might change as they increase production. The marginal cost can be computed using a simple formula: \[ MC(q) = C(q+1) - C(q) \]where:

• \( MC(q) \) is the marginal cost at quantity \( q \)
• \( C(q+1) \) is the total cost of producing \( q+1 \) units
• \( C(q) \) is the total cost of producing \( q \) units

In the original problem, the marginal cost \( MC(500) = 15 \) means that producing the 501st unit costs $15 more than producing 500 units.
Using marginal cost helps businesses decide whether increasing production is financially sound.

Exploring Marginal Revenue

Marginal revenue refers to the additional income generated from selling one extra unit of a good or service. It informs producers about how much revenue will increase with increased production. The calculation for marginal revenue is:\[ MR(q) = R(q+1) - R(q) \]This translates to:

• \( MR(q) \) as the marginal revenue at quantity \( q \)
• \( R(q+1) \) as the revenue for selling \( q+1 \) units
• \( R(q) \) as the revenue for selling \( q \) units

In the exercise, \( MR(500) = 20 \) indicates that the revenue increases by $20 when one more unit is sold, from the 500th to the 501st unit.
Understanding marginal revenue helps firms make better pricing and production decisions.

Revenue Calculation Explained

Revenue calculation is a fundamental aspect of business operations. It represents total earnings from goods or services sold over a period. Businesses calculate revenue by multiplying the price per unit by the number of units sold.The basic formula is:\[ R(q) = p \times q \]where:

• \( R(q) \) denotes revenue from selling \( q \) units
• \( p \) represents the price per unit
• \( q \) is the quantity sold

In the problem, given that revenue at 500 units is $9,400, revenue calculation starts with known values like these to compute profits and make informed business decisions.
Accurate revenue calculation is vital for evaluating business performance and setting financial goals.

Diving into the Cost Function

The cost function illustrates the total cost incurred by a company in producing a particular number of units. It generally represents fixed costs plus variable costs that change with production volumes. The standard cost function is represented as:\[ C(q) = FC + VC(q) \]where:

• \( C(q) \) is the total cost for \( q \) units
• \( FC \) are the fixed costs that do not change regardless of production volume
• \( VC(q) \) is the variable cost that depends on the quantity produced

In our example, \( C(500) = 7200 \) means that the cost of producing 500 units amounts to $7,200 including all relevant expenses.
The cost function is crucial for businesses to plan and control their expenses and decide on pricing strategies.