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The total revenues \(R\) (in millions of dollars) for Krispy Kreme from 2000 through 2007 are shown in the table. $$\begin{array}{|c|c|}\hline \text { Year } & \text { Revenue, } \boldsymbol{R} \\\\\hline 2000 & 300.7 \\\2001 & 394.4 \\ 2002 & 491.5 \\\2003 & 665.6 \\\2004 & 707.8 \\\2005 & 543.4 \\\2006 & 461.2 \\\2007 & 429.3 \\\\\hline\end{array}$$ A model that represents these data is given by \(R=3.0711 t^{4}-42.803 t^{3}+160.59 t^{2}-62.6 t+307\) \(0 \leq t \leq 7,\) where \(t\) represents the year, with \(t=0\) corresponding to 2000. (Source: Krispy Kreme) (a) Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. (b) How well does the model fit the data? (c) Use a graphing utility to approximate any relative extrema of the model over its domain. (d) Use a graphing utility to approximate the intervals over which the revenue for Krispy Kreme was increasing and decreasing over its domain. (e) Use the results of parts (c) and (d) to write a short paragraph about Krispy Kreme's revenue during this time period.

An open box is to be made from a rectangular piece of material, 15 centimeters by 9 centimeters, by cutting equal squares from the corners and turning up the sides. (a) Let \(x\) represent the length of the sides of the squares removed. Draw a diagram showing the squares removed from the original piece of material and the resulting dimensions of the open box. (b) Use the diagram to write the volume \(V\) of the box as a function of \(x\). Determine the domain of the function. (c) Sketch the graph of the function and approximate the dimensions of the box that will yield a maximum volume. (d) Find values of \(x\) such that \(V=56\). Which of these values is a physical impossibility in the construction of the box? Explain.

A company that manufactures bicycles estimates that the profit \(P\) (in dollars) for selling a particular model is given by \(P=-45 x^{3}+2500 x^{2}-275,000, \quad 0 \leq x \leq 50\) where \(x\) is the advertising expense (in tens of thousands of dollars). Using this model, find the smaller of two advertising amounts that will yield a profit of $$\$ 800,000\(.\)

A manufacturer wants to enlarge an existing manufacturing facility such that the total floor area is 1.5 times that of the current facility. The floor area of the current facility is rectangular and measures 250 feet by 160 feet. The manufacturer wants to increase each dimension by the same amount. (a) Write a function that represents the new floor \(\operatorname{area} A\). (b) Find the dimensions of the new floor. (c) Another alternative is to increase the current floor's length by an amount that is twice an increase in the floor's width. The total floor area is 1.5 times that of the current facility. Repeat parts (a) and (b) using these criteria.

A third-degree polynomial function \(f\) has real zeros \(-2, \frac{1}{2},\) and \(3,\) and its leading coefficient is negative. Write an equation for \(f\). Sketch the graph of \(f\). How many different polynomial functions are possible for \(f ?\)

Use the information in the table to answer each question. $$\begin{array}{|c|c|}\hline \text { Interval } & \text { Value of } f(x) \\\\\hline(-\infty,-2) & \text { Positive } \\\\\hline(-2,1) & \text { Negative } \\\\\hline(1,4) & \text { Negative } \\\\\hline(4, \infty) & \text { Positive } \\\\\hline\end{array}$$ (a) What are the three real zeros of the polynomial function \(f ?\) (b) What can be said about the behavior of the graph of \(f\) at \(x=1 ?\) (c) What is the least possible degree of \(f ?\) Explain. Can the degree of \(f\) ever be odd? Explain. (d) Is the leading coefficient of \(f\) positive or negative? Explain. (e) Write an equation for \(f\). (There are many correct answers.) (f) Sketch a graph of the equation you wrote in part (e).