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. Find a model for the revenue from annual sales of blank videocassettes. Give the model as a quadratic trinomial.

Short answer

The model for the revenue from annual sales of blank videocassettes is \(R = -3.15t^2 + 52.92t + 988.72\), where \(R\) is measured in millions of dollars.

Step-by-step solution

Understand the relationship

The revenue \(R\) from annual sales of blank videocassettes is the product of the number of units sold \(B\) and the price per unit \(P\). Hence, \(R = B \times P\).

Substitute the models into the revenue equation

The number of units sold \(B\) is modeled as \(15t + 281\) and the price per unit \(P\) is modeled as \(-0.21t + 3.52\). Substituting these equations into the revenue equation gives \(R = (15t + 281)(-0.21t + 3.52)\)

Multiply the equations

Multiplying the equations results in an expanded quadratic equation: \(R = -3.15t^2 + 52.92t + 988.72\)

Key concepts

Quadratic Trinomial

In algebra, a quadratic trinomial is a polynomial that has the standard form of \(ax^2+bx+c\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). These expressions are called trinomials because they have three terms, and quadratic because the highest power of the variable is two.

When it comes to real-world applications, modeling scenarios with quadratic trinomials allows us to represent various phenomena, such as projectile motion, area of an object, or, as in our textbook example, revenue generation over time. The key features of a quadratic trinomial, such as its vertex and axis of symmetry, can provide valuable insights into the behavior of the quantity being modeled.

In the problem we're examining, the model for revenue is given by a quadratic trinomial that occurs from multiplying a linear equation of units sold (\(B\)) by a linear equation of price per unit (\(P\)). This multiplication expands to a quadratic expression representing the revenue over time (\(t\)). The coefficients of the trinomial hold significant meaning: \(a\) indicates the rate of acceleration or deceleration in revenue, \(b\) shows the linear rate of change, and \(c\) represents the starting value at time \(t=0\).

Understanding quadratic trinomials and their properties, such as how to find their vertex and intercepts, is crucial as it helps in graphing their parabolic shapes and in solving maxima or minima problems in various disciplines.

Linear Equations

Moving on to linear equations, these are fundamental to algebra and represent relationships between two quantities that vary proportionally. A linear equation in one variable has the form \(mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This describes a straight line when graphed on a coordinate plane. The slope indicates the steepness and direction of the line, while the y-intercept is the point where the line crosses the y-axis.

Linear equations are versatile tools for modeling real-world problems where a steady rate of change is present. In our textbook exercise, both the number of units sold (\(B\)) and the price per unit (\(P\)) are described using linear equations. These equations provide simple yet powerful models to predict future sales or price changes based on past data.

To improve comprehension in an educational setting, visual aids like graphs can greatly assist students in understanding the relationship between the variables in a linear equation. By graphing such equations, students can observe the linear behavior and predict outcomes for different values of \(t\), which in this context represents time since 1987.

Algebraic Expressions

Lastly, let's delve into algebraic expressions, which are combinations of numbers, variables, and arithmetic operations (addition, subtraction, multiplication, and division). Unlike equations, algebraic expressions do not contain an equals sign. They are essential building blocks of algebra and are used to represent values and relationships in mathematical problems.

An effective way to handle algebraic expressions is to simplify or manipulate them using algebraic rules like the distributive property, combining like terms, or factoring. These techniques enable students to rewrite complex expressions in a more manageable form or solve for unknowns.

In our textbook exercise, the annual revenue from videocassette sales model combines two algebraic expressions through multiplication to form the quadratic trinomial for revenue. By practicing how to expand and simplify products of algebraic expressions, students gain proficiency in creating models that can lead to more accurate predictions and better understanding of the variables influencing those predictions.

To solidify students' grasp of algebraic expressions, presenting examples in a step-by-step manner, followed by exercises where they can apply similar methods, is often a beneficial teaching strategy. It allows students to actively engage with the content and understand the operations they perform on expressions more meaningfully.