Algebra

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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.

  • Algebra -

    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?

  • Algebra -

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

    27.4*(.961)^t))

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