# Algebra

posted by Katie

The yearly per capita consumption of whole milk in the United States reached a peak of 40 gallons in 1945, at the end of World War II. By 1970 consumption was only 27.4 gallons per person. It has been steadily decreasing since 1970 at a rate of about 3.9% per year.

(a)
Construct an exponential model for per capita whole milk consumption (in gallons) where years since 1970.

1. Reiny

assume an equation of the kind,

C = a e^(kt), where C is the consumption, a is the intital amount , t is the number of years since 1945, and k is a constant.

given:
when t = 0, C = 40,
40 = a e(k(0))
a = 40
so we have:
C = 40 e^(kt)

given : when t = 25 (1970), C = 27.4
27.4 = 40 e(25k)
take ln of both sides
ln 27.4 = ln 40 + 25k
k = (ln 27.4 - ln 40)/25
= -.015133..

C = 40 e^(-.015133.. t) <------ equation #1

or with a base of .961
C = 40 (.961)^(kt)
when t = 25, C = 27.4

27.4 = 40 (.961)^(25k)
log both sides:
log 27.4 = log 40 + 25k(log .961)
(log 27.4 - log 40)/(25log.961) = k
k = .38042

C = 40 (.961)^(.38042t) <----- second version

checking the last one:
if t = 25
C = 40 (.961^9.5105)
= 27.400001306 , pretty good eh?

2. Katie

Thanks for the help! It actually helped it... the answer was

27.4*(.961)^t))

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