Question

In Exercises 56 and \(57,\) use the following information about videocassette sales from 1987 to \(1996,\) where \(t\) is the number of years since 1987 . The annual number of blank videocassettes \(B\) sold in the United States can be modeled by \(B=15 t+281,\) where \(B\) is measured in millions. The wholesale price \(P\) for a videocassette can be modeled by \(P=-0.21 t+3.52,\) where \(P\) is measured in dollars. Describe what happens to revenue during the period from 1987 to 1996 .

Short answer

The revenue from the videocassette sales increases up to a certain year during the period from 1987 to 1996, and then starts to decrease.

Step-by-step solution

Understand the given functions

Two linear functions are given, one represents the number of blank videocassettes sold over time (from 1987 to 1996) and the other represents the wholesale price for a videocassette over the same time period. The equations are \(B=15 t+281\) where \(t\) is the time since 1987 and \(P=-0.21 t+3.52\). The number of videocassettes sold is increasing over time, while the price is decreasing over time.

Compute the revenue function

The revenue function, which we'll call \(R(t)\), can be found by multiplying the number of units sold by the price per unit. In this case, this means multiplying the two given functions. So, \(R(t) = B(t) * P(t)\). When we substitute the given functions in, the result is \(R(t) = (15 t+281)*(-0.21 t+3.52)\).

Simplify the revenue function

To simplify the equation, distribute the terms in the second parentheses to get \(R(t) = ( -3.15 t^2 +382.32 t+989.52)\). This is the required equation for Revenue over the given time period.

Investigate the behavior of the revenue function

The given function for revenue is a quadratic function, which means the revenue function forms a parabola. In this case, since the coefficient of \(t^2\) is negative, the parabola opens downward, indicating that the revenue increases to a certain point and then starts to decrease.

Key concepts

Revenue Function

Imagine you run a business selling a product, and you want to know how your total income—what we call revenue—will change over time. The revenue function in algebra is key to understanding this. It’s simply the product of the number of items sold and the price at which each item is sold.

In the context of the videocassette example, you start with two bits of information: how many videocassettes are sold each year (\(B\text{ in millions}\)), and the price at which they are sold (\(P\text{ in dollars}\)). To find out how the revenue (R) evolves, you multiply these two values together to get the revenue function: \(R(t) = B(t) \cdot P(t)\). After performing this multiplication, the result is an equation that will show you the revenue at any point during the time frame.However, things aren’t constant; the exercise reveals a linear decline in the price of each tape, while the number of tapes sold increases linearly. By creating this revenue function for videocassettes, we get a dynamic look at the business's health. Originally, revenue grows, but due to the decreasing price, there eventually comes a turning point, after which despite selling more tapes, the total revenue will start to drop!

Modeling with Linear Equations

When faced with real-world problems, using modeling with linear equations is like having a mathematical crystal ball. These equations give us a formula to predict future trends based on current data.

In our textbook case, the number of videocassettes sold and their price over time are both modeled as linear equations. A linear equation is recognizable by its straight-line graph and the standard form \(y = mx + b\), where 'm' stands for the slope and 'b' is the y-intercept. In simpler terms, 'm' tells you how quickly the number of cassettes or the price changes over time, while 'b' indicates the starting point.

Using linear functions, we can easily forecast the future—you just plug in the number of years into the equation, and you'll know the expected sales or price. This clarity is essential for businesses to plan for production, pricing strategies, and marketing plans. It turns abstract numbers into a story that reveals the potential future and empowers decision-makers.

Quadratic Functions

As we venture further into the realm of algebra, we find ourselves at the doorstep of quadratic functions. Unlike the straight path of linear functions, quadratic functions create a parabolic curve that can open upwards or downwards on a graph. The standard form of a quadratic equation is \(y = ax^2 + bx + c\).In our case, after finding the revenue function, we discover it's a quadratic equation, which happens to be parabolic and opens downwards due to the negative coefficient of \(t^2\). This shape tells us that the revenue will not just increase or decrease linearly. Instead, it forms a curve where revenue will reach a peak at a particular time before declining.

The 'discriminant' in the quadratic equation, that is \(b^2 - 4ac\), can tell you about the nature of the roots of the equation, but here, our interest is in the visual—the parabola tells us the success story of the videocassettes sales over time. Hence, understanding the nature of quadratic functions is not just about solving an equation, it's about envisioning the ups and downs of a situation, quite literally.