Exponential functions

[Dazed]

Hi, could someone please help me with these 2 problems by finding out the answers to each part. Any help ASAP would be greatly appreciated. Thanks.

1. A table of the number of days that milk will keep when stored at various temperatues is shown in the table.

Temperature - Days
30 - 24
38 - 10
45 - 5
50 - 2
60 - 1
70 - 0


a. Find an exponential model for the data.

I know the formula is f(x)=ab^x
I know b would be the % change but what would a be?

b. If a refrigerator is adjusted to 40F from 37F how much sooner will the milk spoil when stored in this refrigerator?



2.

Data table gives the number of transistors (in millions) in Intel processor chips.

Processor - Year - Number of transistors (millions)
4004 - 1971 - .0023
80286 - 1982 - .134
386DX - 1986 - .275
486DX - 1989- 1.2
Pentium - 1993 - 3.1
Pentium Pro - 1995 - 5.5
Pentium II- 1997 - 7.5
Celeron - 1999 - 19.0
Pentium IV - 2000 - 42.0
Pentium IV - 2002 - 55.0


a. Find an exponential model for the data.

b. According to the model ffound in part a, what is the annual percentage increase in the number of transistors used?

c. Is it reasonable to expect this exponential growth rate to continue beyond the year 2003? explain.



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[Unco]

To find an exponential model:

You would normally graph the data with Days on the x-axis and Temp on the y-axis to see that it looks exponential, but you're told to use this model anyway.

The general equation is
y = Ae^(kx) (we'll rewrite that soon)

k is found by drawing a log(Temp) vs Days graph: k is the gradient of the line of best fit (the data will now be reasonably linear).

A is found by substituting a point (Days, Temp) e.g. (24,30), and the value of k you found earlier into the equation.

If you had worked it out to be, say,
y = 60e^(-0.05x)

as e^(-0.05) = 0.9512...
we have y = 60 * 0.9512^x (which is of form y = a*b^x).

For 1b., you could plug in y=40 and y=37 to get their corresponding values for x (i.e days) and find the difference.

See how you go from there.