# 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?

Jul 29, 2016

Annual consumption in 2000 $\cong 7.46$ 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.

${Q}_{t} = {Q}_{1985} {\left(1 - \frac{0.04}{n}\right)}^{n \left(t - 1985\right)}$ Where n is the number of times per year the decline is computed.

If we assume the process is continious, then:

${Q}_{t} = L i {m}_{\text{n->oo}} {Q}_{1985} {\left(1 - \frac{0.04}{n}\right)}^{n \cdot \left(t - 1985\right)}$

$= {Q}_{1985} \cdot {e}^{-} \left(0.04 \cdot \left(t - 1985\right)\right)$

Therefore, ${Q}_{2000} = {Q}_{1985} \cdot {e}^{-} \left(0.04 \cdot \left(2000 - 1985\right)\right)$

${Q}_{2000} = 13.6 \cdot {e}^{-} \left(0.04 \cdot 15\right)$

$= 13.6 \cdot {e}^{-} 0.6 \cong 7.46$ 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:

${Q}_{2000} = 13.6 \cdot {\left(1 - 0.04\right)}^{15} \cong 7.37$ gallons

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