Question

A manufacturer determines that the demand \(x\) for a product is inversely proportional to the square of the price \(p\). When the price is \(\$ 10\), the demand is 2500\. Find the revenue \(R\) as a function of \(x\) and approximate the change in revenue for a one-unit increase in sales when \(x=3000\). Make a sketch showing \(d R\) and \(\Delta R\).

Short answer

Under the given conditions, the revenue function is \(R(x) = x^3\) and the rate of change of revenue is \(dR = 3x^2\). At \(x = 3000\), \(d R = \$27,000,000\). Therefore, the approximate change in revenue \(\Delta R\) for a one unit increase in sales when \(x=3000\), would be \$27,000,000.

Step-by-step solution

Establish the Relation

From the problem, we know that the demand \(x\) is inversely proportional to the square of the price \(p\). This can be expressed as \(x = \frac{k}{p^2}\) where \(k\) is the constant of proportionality. When \(p = 10\) dollars, \(x = 2500\), we can solve for \(k\). Hence, \(k = x * p^2 = 2500 * 10^2 = 250,000\).

Formulate the Revenue Function

Revenue \(R\) is the product of price \(p\) and demand \(x\), or \(R = p * x\). Using our earlier established relation \(x = \frac{k}{p^2}\), we can express \(R\) as a function of \(x\), i.e, \(R(x) = p*x = \frac{k*x}{p} = \frac{250,000x}{p}\). Now, substituting \(p = \frac{k}{x^2}\) we get \(R(x) = x * \frac{250,000}{p} = 250,000x \cdot \frac{x^2}{250,000} = x^3\).

Calculate the Rate of Change of Revenue

The rate of change of revenue \(R\) with respect to demand \(x\), or \(d R\) is given by the derivative of \(R(x)\) with respect to \(x\). \(dR = \frac{d}{dx}(x^3) = 3x^2\). When \(x = 3000\), this becomes \(dR = 3 * 3000^2 = 27,000,000\). Hence the rate of change of revenue for a one-unit increase in sales, when \(x=3000\), is \$27,000,000.

Approximate Change in Revenue

Now, when there is a slight increase in sales, i.e., when \(x\) changes by \(1\), the approximate change in revenue, \(\Delta R\), is given by \(d R \cdot \Delta x\), or just \(d R\) in this case since \(\Delta x = 1\). So, \(\Delta R = 27,000,000\).\n\nTherefore, an increase of one unit in sales when the demand is at 3000 gives approximately \$27,000,000 increase in revenue.

Sketch the Graph

To sketch \(d R\) and \(\Delta R\), we graph the derivative of the revenue function, \(3x^2\), and note the value at \(x=3000\). The result is a parabolic curve. A line tangent to the curve at \(x = 3000\) represents \(d R\) (also represented as \(27,000,000\)). The value of \(\Delta R\) at that point will be represented by the distance between the tangent line and the curve, just above \(x=3000\).

Key concepts

Inverse Proportionality

Inverse proportionality is a fundamental concept in calculus, particularly in economics, where it explains the relationship between two variables when their product is a constant. In the context of demand and price, if a product’s demand, denoted as \(x\), is inversely proportional to the square of its price \(p\), the relationship can be expressed as \(x = \frac{k}{p^2}\), where \(k\) is a constant. This means, as the price increases, the demand decreases, and vice versa, but the product of the demand and the square of the price remains unchanged.

The concept is critical because it explains how varying one quantity impacts another. For instance, a company may lower the price to increase demand, understanding that there's a predictable, inverse relationship that can be quantified and used for making strategic decisions. The use of inverse proportionality in setting prices can help predict demand and plan for production and supply.

Revenue Function

The revenue function in economics is a crucial tool used to calculate the total revenue generated by the sale of goods or services. It is typically expressed as a function of either price or quantity. In our example, \(R(x)\) represents the revenue function with respect to demand \(x\).

After establishing that demand is inversely proportional to the square of the price, and determining the constant of proportionality, we can express revenue as a function of demand. In this case, the revenue function can be simplified to \(R(x) = x^3\) which elegantly shows that revenue is affected by the cube of the demand. This cubic relation is especially potent, as small changes in demand can lead to large changes in revenue, highlighting the importance of managing demand effectively in a business. The revenue function helps in mapping out potential earnings and strategizing for price adjustments or marketing campaigns.

Rate of Change

The rate of change is a powerful concept that measures how a quantity increases or decreases with respect to another variable. Economists often look at the rate of change to understand how sensitive revenue might be to changes in sales or demand. The derivative of the revenue function with respect to demand, \(dR\), gives us precisely this rate of change of revenue.

For instance, by calculating \(dR = 3x^2\), we find out how revenue changes as demand varies. At a demand level of \(x=3000\), the rate of change is extremely high, indicating that a very small increase in demand can lead to a significantly large increase in revenue. Understanding this rate of change can help businesses predict and manage their income streams more precisely.

Derivative

In calculus, the derivative is a measure of how a function changes as its input changes. It is a fundamental tool that allows economists to analyze the behavior of functions like the revenue function in our example. When we take the derivative of the revenue function \(R(x) = x^3\) with respect to demand \(x\), denoted as \(dR = \frac{d}{dx}(x^3) = 3x^2\), we get a new function that shows the rate of change of revenue for any given level of demand.

The derivative also helps in making approximations, which are often necessary when dealing with complex economic models. For example, the approximate change in revenue for a one-unit increase in sales, \(Delta R\), can be estimated using the rate of change given by the derivative. The derivative, therefore, is not just a mathematical abstraction; it is a practical tool for making better business decisions by quantifying and analyzing changes in economic indicators.