Question

PROBLEM SOLVING An online music store sells about 4000 songs each day when it charges \(1 per song. For each \)0.05 increase in price, about 80 fewer songs per day are sold. Use the verbal model and quadratic function to determine how much the store should charge per song to maximize daily revenue.

Short answer

The store should charge $1.75 per song to maximize its daily revenue.

Step-by-step solution

Define Variables

Let \(x\) be the increase in price in multiples of $0.05, and let \(R(x)\) be the revenue.

Formulate the quadratic equation

The number of songs sold per day with the price increase \(x\) can be represented as \(4000 - 80x\). Therefore, the revenue function will be \(R(x) = (1 + 0.05x)(4000 - 80x)\), as the selling price for each song is the original price of $1 plus the increase \(0.05x\).

Simplify the equation

To solve the equation, first expand the brackets and simplify \(R(x) = 4000 - 80x + 200x -4x^2\), and combine like terms to get \(R(x) = 4000 + 120x - 4x^2 \). This equation is now in standard quadratic form.

Determine the maximum

To find the maximum of this quadratic function, we could either solve the derivative equals to zero or use the symmetry property of a parabola: the maximum is at the vertex which is given by \(- \frac{b}{2a}\). Here \(a=-4\) and \(b=120\). So \(x = - \frac{120}{2*(-4)} = 15\).

Convert back to original units

Since \(x\) is an increase in multiples of $0.05, this corresponds to an actual price increase of $0.05 * 15 = $0.75. So, the price per song yielding maximum revenue is $1 + $0.75 = $1.75.

Key concepts

Revenue Maximization

When thinking about maximizing revenue, it's all about finding the sweet spot where a business can make the most money. In this case, the online music store is attempting to sell songs at a price that maximizes its daily revenue. This involves understanding both how much to charge and how sales might drop when prices go up.

Here we have a scenario where starting at $1 per song, every incremental increase of $0.05 results in selling fewer songs. The function for revenue combines these elements: number of songs sold and price per song. Revenue maximization is achieved when the balance between the price and the number of units sold results in the highest possible total earnings.

This is why we use a quadratic equation. It helps us model the relationship between price increase and sales volume so that we can calculate when revenue hits its peak. The key takeaway is that revenue maximization isn't just about setting the highest price, but finding the optimal price point where demand and pricing meet harmoniously.

Parabolas

A parabola is a U-shaped graph that represents a quadratic function. Every parabola has a highest or lowest point called the vertex. In the context of revenue maximization, understanding the shape of the parabola helps us find the price at which revenue is maximized.

For the quadratic function of the music store problem, the parabola opens downward because the coefficient of the \(x^2\) term is negative. This indicates that there is a maximum point on the graph— the vertex. This vertex represents the point of maximum revenue.

• Parabolas have a symmetrical property which means the two halves of the parabola around the vertex are mirror images of each other.
• By using the vertex, specifically the formula \(-\frac{b}{2a}\), we can effectively determine the point of maximum revenue in the problem.

Vertex Form

The vertex form of a quadratic function is a useful way to find the vertex of a parabola directly from an equation. The standard vertex form is given by:\[f(x) = a(x - h)^2 + k\]Here, \((h, k)\) represents the vertex of the parabola.

In our solution, we did not start with the vertex form but transformed the standard quadratic equation to find the vertex algebraically. This is because the quadratic equation was initially set in standard form and then simplified. By finding the vertex, we calculate the value of \(x\) where revenue is maximized. In this instance, the parabolic function that describes the store's revenue was found through adapting the standard form equation to solve for the vertex.

• The transformation to find the vertex uses the properties of symmetry, making it a straightforward substitution into the formula \(-\frac{b}{2a}\).
• Once found, the vertex tells us exactly how much to increase the price to reach maximum revenue.

Such use of vertex form, even indirectly, helps in making informed economic decisions like setting optimal prices.