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 = 8 corresponding to 2008.(a) Use the regression capabilities of a graphing utility to find a model of the form S = at3 + bt2 + ct + d for the data. (b) Use a graphing utility to plot the data and graph the model. (c) Use calculus and the model to find the time t when sales were increasing at the greatest rate. (d) Do you think the model would be accurate for predicting future sales? Explain 10 11 12 13t89.14S8.17.37.89.211.312.8 12.9

Answered: 10 11 12 13 t 8 9. 14 S 8.1 7.3 7.8 9.2…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

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 = 8 corresponding to 2008.(a) Use the regression capabilities of a graphing utility to find a model of the form S = at³ + bt² + ct + d for the data. (b) Use a graphing utility to plot the data and graph the model. (c) Use calculus and the model to find the time t when sales were increasing at the greatest rate. (d) Do you think the model would be accurate for predicting future sales? Explain

10 11 12 13
t
8
9.
14
S
8.1
7.3
7.8
9.2
11.3
12.8 12.9

Expert Solution

Step 1

Given Information:

The data for the annual sales and the time is given. The value of time for 2008 is 8.

Step 2

(a)

The regression model is; $S = a t^{3} + b t^{2} + c t + d$

The normal equations will be,

$\sum_{} S = a \sum_{} t^{3} + b \sum_{} t^{2} + c \sum_{} t + 7 d . . . (1) \sum_{} t S = a \sum_{} t^{4} + b \sum_{} t^{3} + c \sum_{} t^{2} + d \sum_{} t . . . (2) \sum_{} t^{2} S = a \sum_{} t^{5} + b \sum_{} t^{4} + c \sum_{} t^{3} + d \sum_{} t^{2} . . . (3) \sum_{} t^{3} S = a \sum_{} t^{6} + b \sum_{} t^{5} + c \sum_{} t^{4} + d \sum_{} t^{3} . . . (4)$

The table given below shows the values of $S, t, t^{2}, t^{3}, t^{4}, t^{5}, t^{6}, t S, t^{2} S a n d t^{3} S$ .

S.No.	t	S	$t^{2}$	$t^{3}$	$t^{4}$	$t^{5}$	$t^{6}$	$t S$	$t^{2} S$	$t^{3} S$
1	8	8.1	64	512	4096	32768	262144	64.8	518.4	4147.2
2	9	7.3	81	729	6561	59049	531441	65.7	591.3	5321.7
3	10	7.8	100	1000	10000	100000	1000000	78	780	7800
4	11	9.2	121	1331	14641	161051	1771561	101.2	1113.2	12245.2
5	12	11.3	144	1728	20736	248832	2985984	135.6	1627.2	19526.4
6	13	12.8	169	2197	28561	371293	4826809	166.4	2163.2	28121.6
7	14	12.9	196	2744	38416	537824	7529536	180.6	2528.4	35397.6
Total	$\sum_{} t = 77$	$\sum_{} S = 69.4$	$\sum_{} t^{2} = 875$	$\sum_{} t^{3} = 10241$	$\sum_{} t^{4} = 123011$	$\sum_{} t^{5} = 1510817$	$\sum_{} t^{6} = 18907475$	$\sum_{} t S = 792.3$	$\sum_{} t^{2} S = 9321.7$	$\sum_{} t^{3} S = 112559.7$

Putting the values in the above equations.

$69.4 = 10241 a + 875 b + 77 c + 7 d . . . (5) 792.3 = 123011 a + 10241 b + 875 c + 77 d . . . (6) 9321.7 = 1510817 a + 123011 b + 10241 c + 875 d . . . (7) 112559.7 = 18907475 a + 1510817 b + 123011 c + 10241 d . . . (8)$

Solving the above equations, the value of $a, b, c, a n d d$ is;

$a = - 0.1167, b = 3.9798, c = - 43.356, a n d d = 160.04$

Thus, the regression model is, $S = - 0.1167 t^{3} + 3.9798 t^{2} - 43.356 t + 160.04$ .

Step 3

(b)

Using excel, the graph of the data and the model is given as,

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