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?

Question

1907 views around the world

You can reuse this answer
Creative Commons License

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.