Question

An athletic store sells about 200 pairs of basketball shoes per month when it charges \(\$ 120\) per pair. For each \(\$ 2\) increase in price, the store sells two fewer pairs of shoes. How much should the store charge to maximize monthly revenue? What is the maximum monthly revenue?

Short answer

The store should charge $160 to maximize its monthly revenue. The maximum monthly revenue is $26,400.

Step-by-step solution

Formulate the Function

Define the number of pairs sold (N) as a function of the price increase (∆P): \(N = 200 - ∆P * 2/2\). ∆P is the increase from the initial price, so \(P = 120 + ∆P\). Now, the revenue function \(R = P * N\) can be rewritten as \(R = (120 + ∆P)*(200 - ∆P)\).

Simplify the Function

This function \(R = (120 + ∆P)*(200 - ∆P)\) can be expanded and simplified to \(R = 24000 - 80∆P + ∆P^2. This is a quadratic function, and its graph is a parabola that opens upwards. The function will be maximized at the vertex points of the parabola.

Find the Maximum Price

The ∆P that maximizes this function can be found using the formula \(-b/2a\) for the vertex of a quadratic function, where a and b are the coefficients of ∆P^2 and ∆P respectively. Here, a = 1 and b = -80. So, \(∆P = -(-80) / 2*1 = 40\). So, the price that maximizes the revenue is \(P = 120 + 40 = $160\).

Calculate the Maximum Revenue

Substitute ∆P = 40 into the revenue function \(R = 24000 - 80*40 + 40^2 = 26400.\) So, the maximum monthly revenue is $26,400.

Key concepts

Quadratic Functions

Quadratic functions are mathematical expressions that represent parabolas when graphed on a coordinate axis. They take the general form of
\( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the independent variable. In business applications, quadratic functions often model scenarios where there are diminishing returns or increasing costs, such scenarios that include profit maximization and pricing strategy.

In the context of the exercise, the quadratic function describes the relationship between the change in price for a pair of basketball shoes and the revenue generated from sales. As the price changes, so does the number of shoes sold, and hence the total revenue, which can be depicted as a parabola. Here, the goal is to find the optimal price that maximizes revenue, which occurs at the peak of the parabola.

Vertex Formula

The vertex of a parabola is the highest or lowest point on its graph, and crucially for business decisions, represents the maximum or minimum value of the quadratic function. For a parabola that opens downwards, the vertex represents the maximum value, and conversely for an upward-opening parabola, it represents the minimum value.

The vertex formula arises from calculus and is given as \( -b/(2a) \) when dealing with quadratic functions formulated as \( ax^2 + bx + c \). In our exercise, the formula helped determine the price increase that maximizes the store's revenue. By substituting the values of \( a \) and \( b \) from the revenue function into the vertex formula, one can find the specific change in price that leads to the highest possible revenue, thereby solving a key problem in price optimization strategies.

Price Optimization

Price optimization is the use of mathematical analysis to determine how businesses can adjust their prices to maximize profits. The process includes analyzing data and using models to forecast demand at various price levels, understanding the market, and taking into account cost constraints and inventory levels.

Importance in Business

In the exercise provided, price optimization is pivotal, as selecting the right price for basketball shoes impacts the store's revenue. Through the use of the quadratic function and the vertex formula, the optimal price point is calculated. It's significant in real-world applications that companies must consider several external factors, such as competitor pricing and customer price sensitivity, to fine-tune their optimization models beyond just the theoretical optimum.

Revenue Calculation

Revenue calculation involves determining the total amount of money generated from the sale of goods or services before deducting expenses or costs. It's crucial for businesses to understand their revenue patterns to make informed decisions about production, pricing, and marketing strategies.

In the athletic store's case, the calculation entailed determining the total income from selling basketball shoes at various prices, based on the quadratic revenue function. The maximum revenue was found when this function reached its vertex, indicating the best pricing strategy. This quantitative approach allows businesses to predict financial outcomes and adjust their strategies accordingly.