The manager of a store recorded the annual sales S (in thousands of dollars) of a product over a period of 7 years, as shown in the table, where t is the time in years, with t = 6 corresponding to 2006. t s 6 5.4 7 6.9 8 11.5 9 15.5 10 19.0 11 22.0 12 23.6 a. Use the regression features of a graphing calculator to find a model of the form S = a + bť + ct+d for the data above. (2 points) b. Use a graphing calculator to plot the data and sketch the model. (2 points) c. Use calculus and the model to find the time t when sales were increasing at the greatest rate. (4 points) d. Do you think the model would be accurate for predicting future sales? Explain. (2 points)

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EXPERT ANSWER

a) Putting the values in desmos, we get the estimated model to be the following:

b) The graph of the data is the following:

c) Observe that when the sales was increasing as greatest rate, then

Hence the sales at that time would be

d) No, the model would not be great for predicting future sales, as after some point this model would predict negative sales which is not possible.