Question

equation:
predict the consumption of water bottles in 2021:

Explanation:

Step1: Define exponential model

The exponential regression model is $y = ab^t$, where $a$ is the initial value, $b$ is the growth factor, $t$ is years since 1980, and $y$ is consumption.

Step2: Calculate $a$ (t=0 value)

When $t=0$, $y=2.4$, so $a=2.4$.

Step3: Find growth factor $b$

Use $t=5, y=4.5$:
$4.5 = 2.4b^5$
$b^5 = \frac{4.5}{2.4} = 1.875$
$b = 1.875^{\frac{1}{5}} \approx 1.134$

Step4: Compute t for 2021

2021 - 1980 = 41, so $t=41$.

Step5: Predict consumption

$y = 2.4 \times (1.134)^{41}$
First calculate $(1.134)^{41} \approx e^{41 \times \ln(1.134)} \approx e^{41 \times 0.1253} \approx e^{5.1373} \approx 170.1$
$y \approx 2.4 \times 170.1 \approx 408.2$

Answer:

$\approx 408.2$ gallons per capita