Problem 18 - Applied Calculus

Question

The demand equation for a product is \(p=45-0.01 q\) Write the revenue as a function of \(q\) and find the quantity that maximizes revenue. What price corresponds to this quantity? What is the total revenue at this price?

Step-by-Step Solution

Answer

The quantity that maximizes revenue is 2250 units, the corresponding price is $22.5, and the total revenue is $50,625.

Step 1: Write the Revenue Function

The revenue \(R\) is calculated as the product of the price \(p\) and the quantity \(q\). Given the demand equation \(p = 45 - 0.01q\), we can express revenue as a function of \(q\):\[ R(q) = p \cdot q = (45 - 0.01q) \cdot q = 45q - 0.01q^2 \]

Step 2: Find the Quantity that Maximizes Revenue

To find the quantity \(q\) that maximizes revenue, we need to take the derivative of the revenue function \(R(q) = 45q - 0.01q^2\) with respect to \(q\), and set it equal to zero.\[ \frac{dR}{dq} = 45 - 0.02q = 0 \]Solving for \(q\), we get:\[ 0.02q = 45 \]\[ q = \frac{45}{0.02} = 2250 \]

Step 3: Find the Price Corresponding to the Optimal Quantity

Substitute \(q = 2250\) into the demand equation to find the corresponding price \(p\):\[ p = 45 - 0.01 \times 2250 = 45 - 22.5 = 22.5 \]

Step 4: Calculate the Total Revenue at Optimal Quantity and Price

Using the revenue function \(R(q) = 45q - 0.01q^2\), substitute \(q = 2250\) to calculate total revenue:\[ R(2250) = 45 \times 2250 - 0.01 \times 2250^2 = 101250 - 50625 = 50625 \]

Key Concepts

Demand Equation

To understand how the demand equation impacts revenue maximization, let's first explore what it represents. A demand equation shows the relationship between the price of a product and the quantity demanded by consumers. In this context, we're using the simple linear equation:

\( p = 45 - 0.01q \)

This equation suggests that as more units of product are sold, the lower the price becomes. It's common for higher quantities to push down the price consumers are willing to pay. Demand equations are essential for businesses to understand market behavior and play a significant role in determining pricing strategies. They help forecast how changes in price affect demand, and consequently, potential revenue.

Revenue Function

The revenue function is at the heart of any business looking to maximize its profit. Revenue is calculated by multiplying the price per unit by the quantity of units sold. In our problem, we express revenue as a function of quantity using the demand equation:

\( R(q) = p \cdot q = (45 - 0.01q) \cdot q = 45q - 0.01q^2 \)

This equation gives us the total revenue possible for every level of demand, accounting for the relationship between price and quantity as described by the demand equation. Understanding and calculating the revenue function allows businesses to predict the income generated from different sales volumes, which is crucial for strategic decision-making.

Derivative

Calculating the derivative of a function is a powerful method in mathematics used to find the rate of change. For revenue maximization, we need to take the derivative of the revenue function with respect to quantity because it helps identify the maximum point of the function. For the given revenue function

\( R(q) = 45q - 0.01q^2 \)

we find the derivative by applying basic calculus rules:

\( \frac{dR}{dq} = 45 - 0.02q \)

Setting this derivative equal to zero allows us to find the quantity that maximizes revenue. By solving \( 45 - 0.02q = 0 \), we determine \( q = 2250 \). This process of finding the derivative supports businesses in making informed decisions on optimal production levels to maximize revenue.

Optimized Pricing

Optimized pricing is crucial for maximizing revenue while considering market demand. Once the optimal quantity is found (in this case, \( q = 2250 \) units), we can use the demand equation to find the corresponding price. Substituting the optimal quantity into the demand equation:

\( p = 45 - 0.01 \times 2250 = 22.5 \)

shows that at 2250 units, the market price should be \( 22.5 \) per unit. In practice, optimized pricing not only considers internal factors but also external market conditions. It's an essential tool for businesses to ensure competitiveness, customer satisfaction, and profitability. Calculating the total revenue at this price, using the revenue function, gives practical insights into financial expectations.

\( R(2250) = 50625 \)