# For 1990 through 2010, the sales S (in millions $) of shoes can be modeled by S = -.98x^5 + 24.6x^4 - 210x^3 + 660x^2 - 320x + 1520 where x is the number of years since 1990. Were there any...

For 1990 through 2010, the sales S (in millions $) of shoes can be modeled by

S = -.98x^5 + 24.6x^4 - 210x^3 + 660x^2 - 320x + 1520 where x is the number of years since 1990. Were there any years in which the sales were about 2 billion?

Hint: Let S = 2000 and get the equation = 0, then solve for the x intercepts.

Window: x min = 0 for 1990, x max = 20 for 2010.

During the years _____ and _____. (1996.7 would be 1996)

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### 2 Answers

Let `S=-.98x^5+24.6x^4-210x^3+660x^2-320x+1520` where S is in millions and t is the number of years since 1990.

To find where sales are 2 billion (2000 million) set S=2000 and solve. (To solve means to find the zeros -- in this case we are only interested in real zeros so we look for the x-intercepts.)

`2000=-.98x^5+24.6x^4-210x^3+660x^2-320x+1520`

`-.98x^5+24.6x^4-210x^3+660x^2-320x-480=0`

Note that we are guaranteed at least one real zero as this is a quintic. Also note that there can be 1,3, or 5 real zeros. The graph will approach `+oo` as x decreases without bound, and approach `-oo` as x increases without bound.

Input the equation into a graphing utility. With a little effort, you find a convenient viewing window.

(Here the viewing window is xmin=-2,xmax=15, xscl=1,ymin=-1000,ymax=1000,yscl=500.)

See that there are 4 turning points (places where the graph changes from decreasing to increasing or vice versa); this is the maximum number of turning points for a quintic so from the graph there are 3 real zeros.

The first zero occurs between x=-1 and x=0 -- this does not make sense for our model as we start at the year 1990. (The negative value at 0 is not a problem -- take 2000-500=1500 and we sold 1500000 shoes in 1990.)

So the other zeros are the ones we seek. The first is between x=1 and x=2 and the second between x=10 and x=11.

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The sales of shoes hit approximately 2 billion between 1991 and 1992, and then again between 2000 and 2001.

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S=2000

`2000=-.98x^5+24.6x^4-210x^3+660x^2-320x+1520`

`.98x^5-24.6x^4+210x^3-660x^2+320x+480=0`

**Using graphical method we find in year 1992 and 2000 sales S=2000.A points where red line crosses x-axis are the years resp. 2 and 10 approximate**