How do you write an exponential function to model each situation & solve given whole milk consumption in the U.S. has decreased by 4% annually since 1985. Each person consumed 13.6 gallons of whole milk in 1985. Predict whole milk consumption in 2000?

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Answer 1 · 2016-07-29T01:35:28

Annual consumption in 2000 #~= 7.46# gallons (Continuous Model)

Let #Q_t# be the annual consumption in year #t#

We are told that the annual consumption decreases by 4% each year since 1985 and that annual consumption in that year was 13.6 gallons.

Therefore #Q_1985 = 13.6# and the rate of decline is #4% = 0.04# p.a.

#Q_t = Q_1985 (1-0.04/n)^(n(t-1985))# Where n is the number of times per year the decline is computed.

If we assume the process is continious, then:

#Q_t = Lim_"n->oo" Q_1985 (1-0.04/n)^(n * (t-1985))#

#= Q_1985 * e^-(0.04 * (t-1985))#

Therefore, #Q_2000 = Q_1985 * e^-(0.04 * (2000-1985))#

#Q_2000 = 13.6 * e^-(0.04 * 15)#

#= 13.6 * e^-0.6 ~= 7.46# gallons

Note that if the process were discrete and the decline was computed once per year (i.e #n=1# ) the model would result in:

#Q_2000 = 13.6 * (1-0.04)^15 ~= 7.37# gallons

The more frequently the decline is computed each year (i.e. the greater #n#) the closer the result will approach the continious model.