Question

You decide to form a partnership with another business. Your business determines that the demand \(x\) for your product is inversely proportional to the square of the price for \(x \geq 5\). (a) The price is \(\$ 1000\) and the demand is 16 units. Find the demand function. (b) Your partner determines that the product costs \(\$ 250\) per unit and the fixed cost is \(\$ 10,000\). Find the cost function. (c) Find the profit function and use a graphing utility to graph it. From the graph, what price would you negotiate with your partner for this product? Explain your reasoning.

Short answer

The demand function is \(x = 16/p^2\). The cost function is \(C(x) = 10000 + 250x\). The profit function is \(P(p) = 16p/p^2 - 10000 - (4000/p^2)\), which should be graphed to negotiate a price with your partner.

Step-by-step solution

Formulate the demand function

Because demand \(x\) is inversely proportional to the square of the price \(p\), it can be represented as \(x = kp^2\), where \(k\) is a constant of proportionality. Given a price of \$1000 and demand of 16 units, this constant can be obtained by substituting these values into the equation: \(k = 16/1000^2\). This gives us the demand function \(x = 16/p^2\).

Define the cost function

The cost function consists of fixed costs and the costs proportionate to the quantity of units produced. Given a per unit cost of \$250 and a fixed cost of \$10,000, the cost function \(C(x)\) can be written as \(C(x) = 10000 + 250x\).

Determine the profit function

Profit is defined as Revenue (which is Price multiplied by Demand) minus Cost. This gives us the profit function \(P(p) = px - C(x) = p(16/p^2) - (10000 + 250(16/p^2))\). After simplification we get \(P(p) = 16p/p^2 - 10000 - (4000/p^2)\).

Graph and interpret the profit function

The profit function \(P(p)\) should be plotted on a graphing utility, with price \(p\) on the x-axis and profit on the y-axis. Observe where \(P(p)\) is at its peak—that price would maximize your profit and would be the price you should negotiate with your partner.

Key concepts

Demand Function

In business economics, the demand function helps to understand how many units of a product will be demanded at different price levels. It illustrates the relationship between the price of the product and the quantity demanded. If the demand for a product is inversely proportional to the square of its price, as in this exercise, it implies that a higher price will lead to a lower demand, and a lower price will increase demand.
We can express this relationship mathematically by using the formula for inverse proportionality: \( x = \frac{k}{p^2} \), where \( x \) is the demand, \( p \) is the price, and \( k \) is the constant of proportionality. In our example, given a price of \( \$1000 \) and a demand of 16 units, the constant \( k \) can be calculated as \( k = 16 \times 1000^2 \). This calculation results in the demand function \( x = \frac{16}{p^2} \).
The demand function is crucial for businesses as it helps to determine the expected sales at any given price level. This knowledge can guide pricing strategies and help maximize both revenue and profit.

Cost Function

Understanding the cost function is essential for any business as it shows how total cost changes with the level of output. A cost function represents total cost incurred by producing a certain number of units, including both fixed and variable costs. Fixed costs, like the \( \\(10,000 \) mentioned in the exercise, are expenses that do not change with the level of output, such as rent and salaries.
Variable costs, in this case the \( \\)250 \) per unit, change with the number of units produced. By combining these, we define the cost function \( C(x) \) as \( C(x) = 10000 + 250x \). Here, \( x \) is the number of units. The cost function is vital to determine how cost-efficient production is and to ensure expenses remain manageable.
Businesses use the cost function to make informed decisions about scaling operations and to identify potential areas for cost savings, ensuring that the operations are economically viable.

Inverse Proportionality

Inverse proportionality is a mathematical relationship where a decrease in one quantity results in an increase in another and vice versa. In our exercise, the demand \( x \) is inversely proportional to the square of the price \( p \), meaning that as the price increases, the demand decreases. This is represented by the equation \( x = \frac{k}{p^2} \).
This concept is critical in pricing strategies since a small increase in price could significantly reduce demand if the product is sensitive to price changes. Inversely, a price reduction can lead to a substantial increase in demand if the product is perceived as more affordable.
Understanding inverse proportionality allows businesses to predict and model how changes in price will affect demand, allowing for more dynamic and responsive decision-making regarding pricing.

Price Optimization

Price optimization involves finding the most profitable pricing strategy by balancing sales volume with profit margins. The goal is to determine the price point where profits are maximized. This requires understanding both the demand and cost functions, as well as market conditions.
The profit function in this exercise, given by \( P(p) = 16\frac{p}{p^2} - 10000 - \frac{4000}{p^2} \), combines these elements by showing profit as a function of price. By graphing this function, you can visualize where the highest profit occurs, which helps identify the ideal price point.
Achieving optimal pricing ensures maximum profitability while maintaining competitive pricing in the market. Businesses utilize price optimization regularly to adjust prices based on economic conditions, competition, and consumer behavior, ensuring that they maximize revenue while meeting market demands.