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?

1 Answer
Jul 29, 2016

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

Explanation:

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.