Question

A store manager estimates that the demand for an energy drink decreases with increasing price according to the function \(d(p)=\frac{100}{p^{2}+1},\) which means that at price \(p\) (in dollars), \(d(p)\) units can be sold. The revenue generated at price \(p\) is \(R(p)=p \cdot d(p)\) (price multiplied by number of units). a. Find and graph the revenue function. b. Find and graph the marginal revenue \(R^{\prime}(p)\). c. From the graphs of \(R\) and \(R^{\prime}\), estimate the price that should be charged to maximize the revenue.

Short answer

Based on the following step by step solution, create a short answer question: Question: Given the demand function for an energy drink, \(d(p) =\frac{100}{p^{2}+1}\), and the revenue function, \(R(p) = p \cdot d(p)\), find the simplified revenue function and the simplified marginal revenue function. Estimate the price that maximizes the revenue from the graphs of R and R'. Answer: The simplified revenue function is R(p) = \(\frac{100p}{p^{2}+1}\), and the simplified marginal revenue function is R'(p) = \(\frac{100 - 100p^2}{(p^2+1)^2}\). To estimate the price that maximizes the revenue, analyze the graphs of R(p) and R'(p) and find the value of p where the marginal revenue is zero.

Step-by-step solution

Calculate the revenue function R(p)

To find the revenue function, we will multiply the price p by the demand function. R(p) = p * d(p) R(p) = p * (\frac{100}{p^{2}+1})

Simplify the revenue function

R(p) = \frac{100p}{p^{2}+1} Now, the revenue function R(p) is given by: R(p) = \frac{100p}{p^{2}+1} We can graph this function using graphing software or by plotting points. #b. Find and graph the marginal revenue R'(p).#

Calculate the derivative of R(p)

To find the marginal revenue, we will differentiate R(p) with respect to p. We can use the quotient rule, which states that if we have a function in the form of \(\frac{f(x)}{g(x)}\), then its derivative is given by \(\frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}\). f(p) = 100p g(p) = p^2 + 1 f'(p) = 100 g'(p) = 2p Now, apply the quotient rule: R'(p) = \frac{f'(p)g(p) - f(p)g'(p)}{(g(p))^2} R'(p) = \frac{100(p^2+1) - 100p(2p)}{(p^2+1)^2}

Simplify the marginal revenue function

R'(p) = \frac{100(p^2+1) - 200p^2}{(p^2+1)^2} R'(p) = \frac{100p^2+100-200p^2}{(p^2+1)^2} R'(p) = \frac{100 - 100p^2}{(p^2+1)^2} Now, the marginal revenue function R'(p) is given by: R'(p) = \frac{100 - 100p^2}{(p^2+1)^2} We can graph this function using graphing software or by plotting points. #c. Estimate the price that maximizes the revenue from the graphs of R and R'.#

Find the value of p when the marginal revenue is zero

To estimate the price that maximizes the revenue, we can analyze the graph of R(p) and R'(p). At the maximum of the revenue function, the slope of the function will be zero, meaning that the marginal revenue will be zero. From the graph or by analysing the function, we can find an estimate of the price at which the marginal revenue is zero.

Estimate the maximum revenue price

Based on the graph or by analyzing the function itself, we can estimate the value of p that makes marginal revenue equal to zero. This value of p gives us the price that should be charged to maximize the revenue.

Key concepts

Revenue Function

Understanding the revenue function is fundamental in business and economics. In a given market, the revenue function expresses how much money a company can make based on the price of the product and the quantity sold. In our exercise, the revenue function is represented as \( R(p) = p \times d(p) \), where \( p \) stands for the price level and \( d(p) \) is the demand function. To find the revenue function, we multiply the price by the demand for energy drinks at that price level, represented by the formula \( R(p) = \frac{100p}{p^{2}+1} \).

Graphing the revenue function helps visualize how changing the product price affects total revenue. A well-plotted revenue curve can reveal the most advantageous pricing point—where raising or lowering the price could potentially reduce total revenue.

Demand Function Calculus

Demand functions represent the relationship between the product's price and the quantity of the product consumers are willing to purchase. In calculus, the demand function is usually a function of price, symbolized as \( d(p) \). For the energy drink example, the demand function is \( d(p) = \frac{100}{p^{2}+1} \).

This specific form suggests that as the price increases, the quantity demanded decreases, which aligns with the law of demand in economics. Utilizing calculus in this context allows us to explore changes in demand with respect to minute changes in price, ultimately aiding in pricing strategy and revenue maximization efforts.

Derivative Quotient Rule

The derivative quotient rule is a calculus technique used to find the derivative of a function that is the quotient of two other functions. It's expressed as \( \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} \), where \( f(x) \) and \( g(x) \) are the numerator and denominator functions, respectively. To calculate the marginal revenue \( R'(p) \), which is the derivative of the revenue function with respect to price, we apply this rule.

In our energy drink case, the revenue function \( R(p) \) constitutes such a quotient, where \( f(p) = 100p \) and \( g(p) = p^2 + 1 \). By taking the derivatives \( f'(p) = 100 \) and \( g'(p) = 2p \), and applying them to the quotient rule, we derive the marginal revenue function, allowing us to analyze how small changes in price impact the revenue incrementally.

Maximizing Revenue

The ultimate goal in pricing strategy is often to maximize revenue. To determine the price that achieves this, we look at the marginal revenue, \( R'(p) \), which, when graphed, depicts how revenue changes with respect to price. We aim to find the price for which the marginal revenue is zero because it indicates that a further increase in price will start to decrease the revenue.

By graphing or analyzing the marginal revenue function, we can find where \( R'(p) = 0 \), and hence estimate the optimal price point. Although the estimate is not exact, it provides a valuable starting point for setting a price that will maximize the revenue for the energy drinks in this scenario.