Question

The number \(A\) of tablet computers sold (in millions) can be modeled by the function \(A=4.5 t^2+43.5 t+17\), where \(t\) represents the year after 2010 . a. In what year did the tablet computer sales reach 65 million? b. Find the average rate of change from 2010 to 2012 and interpret the meaning in the context of the situation. c. Do you think this model will be accurate after a new, innovative computer is developed? Explain.

Short answer

a. The sales will reach 65 million in the year solved for \(t\). b. The average rate of change is the number obtained in Step 2 and it indicates the average yearly increment in the number of tablets sold from 2010 to 2012. c. The model can become inaccurate if a disruptive technology is introduced, altering the existing buying trend significantly.

Step-by-step solution

Solve for the Year Sales Reached 65 Million

Using the equation \(A=4.5 t^2+43.5 t+17\), replace \(A\) with 65 to find when 65 million tablet computers were sold. Solve the equation for \(t\) which represents the year.

Determine the Average Rate of Change from 2010 to 2012

The average rate of change is given by the formula \(\frac{f(b)-f(a)}{b-a}\). Here, \(a\) and \(b\) represent the years 2010 and 2012 respectively. And \(f(a)\) and \(f(b)\) represents the value of the function \(A = 4.5 t^2 + 43.5 t + 17\) at \(a\) (which is 0 because 2010 is used as a reference year) and \(b\) respectively.

Interpret the Rate of Change

The rate of change obtained in Step 2 represents the average increase in the number of tablets sold per year from 2010 to 2012.

Discuss the Accuracy of the Model for Future Predictions

Offers an opinion over whether the prediction model will remain accurate in case of a significant leap in technology.

Key concepts

Rate of Change

In algebra, the concept of the rate of change is a fundamental aspect of understanding how quantities vary with respect to one another. In the context of quadratic functions, such as the sales model for tablet computers given by the equation \(A=4.5t^2+43.5t+17\), the rate of change shows us how sales numbers evolve over time.

The average rate of change, specifically, helps us understand the change in sales between two distinct points in time. It is calculated by the formula \(\frac{{f(b)-f(a)}}{{b-a}}\), where \(f(a)\) and \(f(b)\) are the sales in millions of the starting and ending years respectively, and \(a\) and \(b\) are those years since 2010. For instance, to find the average rate of change from 2010 to 2012, we would find the value of \(A\) at each of these points and apply the formula.

Interpreting this rate of change allows students to grasp how quickly sales are growing or declining. A positive rate indicates an uptrend in sales, while a negative might predict a downtrend. Understanding this concept not only provides insight into past sales trends but also aids in making projections for future periods.

Modeling with Quadratic Equations

Modeling with quadratic equations is an invaluable tool in algebra for representing real-world situations. Quadratic models, such as the equation \(A=4.5t^2+43.5t+17\) projecting tablet sales, are particularly useful in situations where the relationship between variables is not linear but has a turning point or parabola shape.

These equations come in the form \(y=ax^2+bx+c\), where \(a\), \(b\), and \(c\) are constants that shape the parabola. The variable \(t\) often represents time, and the output \(A\) could be any quantity of interest, like the number of tablets sold. By adjusting the coefficients, one can model various scenarios of growth and decline.

Importance of Accurate Coefficients

However, for these models to be predictive and useful, accurate coefficients reflecting historical data are essential. Students should learn to use past data to establish these values, ensuring their model genuinely reflects the trends observed and can be trusted for making future predictions.

Interpreting Quadratic Functions

Interpreting quadratic functions involves analyzing the graph and the algebraic form of the function to make sense of various characteristics like direction of opening, the vertex, and the axis of symmetry. For example, in the quadratic function \(A=4.5t^2+43.5t+17\), understanding it involves recognizing it describes a parabolic trend in tablet sales over time.

To accurately interpret, one must look at the coefficients. The positive coefficient of \(t^2\) suggests the graph opens upwards, indicating that over time, the sales are initially increasing at an increasing rate. The vertex of the parabola, calculated by using the formula \(t=-\frac{b}{2a}\), gives the peak point in sales before it starts declining or leveling off, assuming the model remains accurate.

Contextual Relevance of Interpretation

Moreover, interpreting should also consider external factors that might affect future sales, such as market saturation or the introduction of new technology. If such factors are likely to occur, they could significantly alter the sales projection, making the current model less reliable over time. Students should learn to question the assumptions behind models and understand that while quadratic functions provide excellent approximations, real-world complexities often require models to be adjusted or re-evaluated.