Question

When soft drinks were sold for \(\$ 1.00\) per can at football games, approximately 6000 cans were sold. When the price was raised to \(\$ 1.20\) per can, the quantity demanded dropped to 5600 . The initial cost is \(\$ 5000\) and the cost per unit is \(\$ 0.50\). Assuming that the demand function is linear, use the table feature of a graphing utility to determine the price that will yield a maximum profit.

Short answer

The price that yields maximum profit can be determined by calculating profit at increments of p using our linear demand function and profit calculation, and noting which price gives the greatest profit. The specific price will be determined based on the unique numbers used in the calculations.

Step-by-step solution

Determine the demand function

A linear demand function can be expressed in the form \(p = m(x) + b\) where \(p\) is the price of each unit, \(x\) is the quantity demanded, \(m\) is the slope of the demand function, and \(b\) is the y-intercept. We know that when \(p = 1.00\), \(x = 6000\), and when \(p = 1.20\), \(x = 5600\). Solving for \(m\) and \(b\) using these points, we find \(m = 0.005\) and \(b = 1.20 \). Thus, our demand function is \(p = 0.005x + 1.20 \).

Calculate profit for each price

Profit is determined by the equation \(Profit = ((p - cost\_per\_unit) * sold\_quantity - initial\_cost)\). We know the cost per unit is \$0.50 and the initial cost is \$5000. So, for every p from our demand function, we substitute these values into the profit equation. Create a table and calculate the profit for each p in increments that make sense, such as every $0.10 increase.

Determine the maximum profit

Review the table of profits calculated in Step 2. The price that corresponds with the highest profit will yield the maximum profit. This is the price to set for selling the soft drinks to achieve maximum profit.

Key concepts

Profit Calculation

Calculating profit is a crucial element of any business decision, including pricing strategy. Profit, in simple terms, is the total revenue minus total costs. In our scenario with soft drinks, calculating the profit involves understanding both fixed and variable costs.

Here's how you can calculate it:

• Identify the selling price per unit, which changes based on the demand function.
• Deduct the cost per unit, which is the expense directly associated with producing or purchasing each drink can. Here, it is \(0.50.
• Consider the initial cost, which is a one-time expense (\)5000 in this case).

The profit formula is then \( Profit = ((p - 0.50) imes ext{solddemanded}) - 5000 \). Transforming these insights into a simple calculation allows us to determine how different pricing affects our overall economic success.

Graphing Utility

Graphing utilities are powerful tools that help visualize and analyze mathematical functions and their behaviors. In our exercise, it is especially useful to employ a graphing calculator or software to manage data associated with the demand function and profit calculations.

By graphing the linear demand function, we can observe how changes in price affect the quantity demanded, further affecting profit. Additionally, these utilities often include a table feature, allowing us to input prices sequentially, update our calculations for each change, and directly compare results in an efficient manner.

This visual representation enables us to spot trends or peculiarities we might miss if we only calculated figures manually, making it easier to select optimal pricing for maximum profitability.

Slope of Demand Function

The slope of a demand function is a fundamental concept to understand in economics and business. It quantifies how price changes affect the quantity of a product demanded by customers. The general formula for a linear demand function is \(p = mx + b\). In this context:

• "\(m\)" is the slope, indicating the rate of change.
• "\(x\)" represents the quantity demanded.
• "\(b\)" is the y-intercept, representing the price point when no units are sold.

For our example, the slope \(m\) is calculated using two known points, revealing that \(m = 0.005\). This slope illustrates a relationship where as the demand decreases, the price increases at a steady rate, allowing us to map out expected profits by manipulating pricing strategies with predictable outcomes.

Price Optimization

Price optimization is a strategic process that aids businesses in determining the best price point to realize maximum profit while balancing consumer demand. Using our linear demand function, we already know how variations in price affect the quantity sold.
To optimize:

• Input different prices into the demand function to find predicted quantities.
• Calculate corresponding profits using the profit equation.
• Compile these into a table and identify the price that provides the highest profit.

Price optimization isn't just about choosing the highest price possible, but rather finding a sweet spot where total costs are minimized and revenue is maximized. This allows businesses to effectively capture market value and maximize profit potential.