Problem 45 - Intermediate Algebra

Question

Use quadratic functions. A movie rental company has 1000 subscribers, and each pays \(\$ 15\) per month. On the basis of a survey, company managers feel that for each decrease of \(\$ 0.25\) on the monthly rate, they could obtain 20 additional subscribers. At what rate will maximum revenue be obtained and how many subscribers will it take at that rate?

Step-by-Step Solution

Answer

The maximum revenue is achieved at a rate of $13.75 with 1100 subscribers.

Step 1: Define the Revenue Function

The revenue function is defined as the product of the price per subscriber and the number of subscribers. Let the decrease in price per subscriber be \( x \). The new price becomes \( 15 - 0.25x \) and the new number of subscribers becomes \( 1000 + 20x \). Therefore, the revenue function \( R(x) \) is: \[ R(x) = (15 - 0.25x)(1000 + 20x) \]

Step 2: Expand the Revenue Function

Now, expand the revenue function by multiplying the two algebraic expressions. This gives: \[ R(x) = 15(1000 + 20x) - 0.25x(1000 + 20x) \] Simplify each term:- The first term is \( 15000 + 300x \).- The second term is \( -250x - 5x^2 \).Combine these to get:\[ R(x) = 15000 + 300x - 250x - 5x^2 \] Simplify by combining like terms:\[ R(x) = 15000 + 50x - 5x^2 \]

Step 3: Calculate the Vertex of the Quadratic Function

To find the maximum revenue, calculate the vertex of the parabola represented by the quadratic function \( R(x) = 15000 + 50x - 5x^2 \). The vertex \( x \) value is given by the formula \( x = -\frac{b}{2a} \), where \( a = -5 \) and \( b = 50 \):\[ x = -\frac{50}{2(-5)} = 5 \]

Step 4: Determine the Maximum Revenue and Rate

Substitute \( x = 5 \) back into the expressions for price and number of subscribers to determine the maximum revenue:- New price per subscriber: \( 15 - 0.25 \times 5 = \\(13.75 \)- Number of subscribers: \( 1000 + 20 \times 5 = 1100 \)Thus, the maximum revenue is achieved at a monthly rate of \( \\)13.75 \) with 1100 subscribers.

Key Concepts

Revenue Function

The revenue function is a mathematical expression that helps businesses calculate their total earnings based on the price per unit and the number of units sold. In the context of a subscription-based model, like a movie rental service, this function is used to find out how much money the company can make from its subscribers.

In our exercise, we defined the revenue function as the product of the subscription price and the number of subscribers. Let's break down the components:

• The starting price for a subscriber is \(15 per month.
• The company considers reducing the price by \)0.25 for each additional x subscribers.
• For every $0.25 price reduction, 20 more subscribers join.
• The new price for the monthly subscription is represented as \( 15 - 0.25x \).
• The updated total number of subscribers is \( 1000 + 20x \).

By combining these elements, the revenue function \( R(x) \) represents the total revenue as \( R(x) = (15 - 0.25x)(1000 + 20x) \). This formula captures the relationship between price changes and subscriber count.

Vertex of a Parabola

To find out the point at which the revenue is maximized, we need to calculate the vertex of the parabola represented by the revenue function. A parabola is a symmetrical curve, and its vertex represents either a maximum or minimum point.

In our quadratic revenue function \( R(x) = 15000 + 50x - 5x^2 \), the graph of the equation forms a parabola. We want to find the maximum revenue, so we need to identify the vertex of this parabola.

• The standard formula to find the vertex \( x \) value for a parabola of the form \( ax^2 + bx + c \) is \( x = -\frac{b}{2a} \).
• Here, \( a = -5 \) and \( b = 50 \).
• Substituting these values, we get \( x = -\frac{50}{2(-5)} = 5 \).

By finding \( x = 5 \), we determine the point of maximum revenue in relation to the changes in the subscription fee and total subscribers.

Maximum Revenue

With the vertex calculation completed, finding the maximum revenue involves plugging the vertex value back into the original problem to determine actual numbers.

At \( x = 5 \):

• The new price per subscriber becomes \( 15 - 0.25 \times 5 = 13.75 \).
• The total number of subscribers updates to \( 1000 + 20 \times 5 = 1100 \).

The maximum revenue occurs when these values and their interactions reflect a balance. The calculated revenue at this point is a direct testament to the financial strategy optimized through the vertex of the revenue function. This approach demonstrates the efficiency of reducing prices modestly while allowing subscriber count to increase.

Subscribers

Subscribers are crucial to the revenue potential of a subscription-based model. Each additional subscriber contributes directly to the total revenue, and thus, understanding how to increase the subscriber base efficiently is key.

In practice, this exercise shows how adjusting the subscription price impacts both the number of subscribers and the total revenue:

• Starting with a subscriber base of 1000, any decrease in price (by steps of $0.25) attracts 20 more subscribers each time.
• This leads to a nonlinear relationship between price changes and subscriber count due to the quadratic nature of the function.
• Reaching a sweet spot where the revenue is maximized (at 1100 subscribers in this scenario) is vital for profitability.

By strategically using this data, companies can make informed decisions on pricing strategies that best influence subscriber behaviors and business growth.