Problem 24 - Applied Calculus

Question

The demand equation for a quantity \(q\) of a product at price \(p\), in dollars, is \(p=-5 q+4000\). Companies producing the product report the cost, \(C\), in dollars, to produce a quantity \(q\) is \(C=6 q+5\) dollars. (a) Express a company's profit, in dollars, as a function of \(q\). (b) What production level earns the company the largest profit? (c) What is the largest profit possible?

Step-by-Step Solution

Answer

The maximum profit occurs at a production level of 399.4 units with a profit of about $798,010.50.

Step 1: Understand Profit Function

Profit is calculated as Revenue minus Cost. Revenue can be found from the demand equation by multiplying price, \(p\), with quantity \(q\). The profit function, \(P(q)\), is thus \(P(q) = pq - C\).

Step 2: Express Revenue as Function of q

From the demand equation \(p = -5q + 4000\), calculate Revenue \(R(q)\) as \((p \times q) = (-5q + 4000)q\). Simplifying this gives \(R(q) = -5q^2 + 4000q\).

Step 3: Calculate Profit Function

The profit function \(P(q)\) is \(R(q) - C(q)\), where \(C(q)\) is the cost function \(6q + 5\). Substituting these into the formula, we get:\[P(q) = (-5q^2 + 4000q) - (6q + 5)\] Simplifying, \[P(q) = -5q^2 + 3994q - 5\].

Step 4: Find Maximum Profit Production Level

To find the production level that maximizes profit, find the vertex of the parabola given by the profit function. The vertex occurs at \(q = -\frac{b}{2a}\) for a quadratic \(ax^2 + bx + c\). Here, \(a = -5\), \(b = 3994\). Thus, \(q = -\frac{3994}{2(-5)} = 399.4\).

Step 5: Verify and Calculate Largest Profit

Substitute \(q = 399.4\) back into the profit function \(P(q) = -5q^2 + 3994q - 5\) to find the maximum profit:\[P(399.4) = -5(399.4)^2 + 3994(399.4) - 5\] The calculated profit is approximately \$798010.5.

Key Concepts

Demand Equation

The demand equation is a fundamental concept that links the price of a product with the quantity demanded by consumers. For this exercise, the demand equation is given as \( p = -5q + 4000 \). Here, \( p \) represents the price in dollars, and \( q \) is the quantity of the product. This equation tells us how much consumers are willing to pay for a given quantity of goods.

• The negative coefficient (-5) indicates that price decreases as quantity increases, reflecting a typical downward-sloping demand curve.
• If no goods are produced \((q = 0)\), the price is \$4000, which is the intercept of the equation.

Understanding the demand equation is crucial, as it forms the basis for calculating the revenue from sales, which in turn affects profit calculations.

Cost Function

The cost function provides insight into the total cost of producing a certain quantity of goods. In this exercise, the cost function is expressed as \( C = 6q + 5 \). Here, \( C \) represents the cost in dollars, and \( q \) is the quantity produced.

• The term \(6q\) indicates the variable cost per unit, meaning for each unit produced, the cost increases by \$6.
• The constant term \(5\) represents the fixed cost of production that occurs regardless of quantity.

By understanding the cost function, companies can determine their expenses at different production levels, which is essential for setting prices and forecasting profitability.

Revenue Function

Revenue is generated by selling products, and it can be computed by multiplying the price \( p \) by the quantity \( q \). From our demand equation \( p = -5q + 4000 \), the revenue function \( R(q) \) becomes \( R(q) = (-5q + 4000)q \). Simplified, this gives \( R(q) = -5q^2 + 4000q \).

• The term \(-5q^2\) implies that the revenue has a quadratic relationship with quantity, forming a parabolic curve.
• The term \(4000q\) represents the initial linear growth part of revenue as more units are sold.

Understanding the revenue function is key to deriving the profit function, as profit is calculated by subtracting costs from revenue.

Maximizing Profit

Maximizing profit is a primary objective for businesses. Profit is calculated as revenue minus cost, and the profit function \( P(q) \) is \( P(q) = (-5q^2 + 4000q) - (6q + 5) \), which simplifies to \( P(q) = -5q^2 + 3994q - 5 \).
To find the quantity \( q \) that maximizes profit, we use the vertex formula for a quadratic function \( q = -\frac{b}{2a} \), where \( a = -5 \) and \( b = 3994 \). This results in a production level \( q = 399.4 \), which is where profit is maximized.
Substituting \( q = 399.4 \) into the profit function calculates the maximum profit, approximately \$798,010.5.

• Finding optimal production levels ensures that resources are efficiently used to generate the highest possible profit.
• Understanding the profit maximization process involves analyzing mathematical models to improve business strategies.