- The cubic model for the data is \(V(t) = 0.017t^3 - 0.428t^2 + 4.429t + 14.293\).
- Using synthetic substitution to evaluate \(V(12)\), we calculate the remainder to be \(35.185\).
- Rounding to the nearest dollar, the estimated value of the stock at week 12 is \(\boxed{\$35}\).
Explanation
- Analysis of Data and Objectives
We are tasked with modeling the value of a stock, \(V\), as a function of time, \(t\) (in weeks), given a set of data points. The data points are (1, 18), (3, 25), (4, 26), (6, 29), (8, 32), (9, 29), (10, 35), and (15, 42). We need to determine a cubic model of the form \(V(t) = at^3 + bt^2 + ct + d\) and then use synthetic substitution to estimate the stock value at \(t=12\).
- Developing the Cubic Model
Using cubic regression on the provided data points, we calculate the best-fit coefficients for the model \(V(t) = at^3 + bt^2 + ct + d\). Rounding each coefficient to the nearest thousandth, we find the model:
\[V(t) = 0.017t^3 - 0.428t^2 + 4.429t + 14.293\]
This cubic function represents the trend of the stock value over the 15-week period following the restructuring.
- Synthetic Substitution at \(t=12\)
To find the stock value at \(t=12\), we use synthetic substitution with the cubic model and the value 12 as our divisor. The synthetic division table is structured as follows:
\[ \begin{array}{c|rrrr} 12 & 0.017 & -0.428 & 4.429 & 14.293 \ & & 0.204 & -2.688 & 20.892 \\ & 0.017 & -0.224 & 1.741 & 35.185 \end{array} \]
Starting with the leading coefficient 0.017, we multiply by 12 to get 0.204. Adding this to the next coefficient (-0.428) yields -0.224. Multiplying -0.224 by 12 gives -2.688, and adding this to 4.429 results in 1.741. Finally, multiplying 1.741 by 12 results in 20.892, which, when added to 14.293, gives the remainder 35.185. This remainder represents the value of the function at \(t=12\). Rounding to the nearest dollar, the estimated stock value is \(\$35\).
Examples
Cubic modeling is an essential tool in financial forecasting and business analysis. By fitting a cubic curve to historical data, analysts can project future trends and estimate potential outcomes, such as stock performance, market growth, or consumer behavior. Synthetic substitution provides an efficient numerical method to evaluate these models, allowing business leaders to quickly estimate values at specific future time intervals, aiding in informed decision-making and strategic planning.