To analyze the model given, \( m(t) = 16.5(0.959)^t \), let's break down the components.
Rate of Decay
The equation represents an exponential decay model, where:
- \( m(t) \) is the consumption of milk in gallons per person,
- \( 16.5 \) is the initial amount of milk consumed per person in the year 2000 (when \( t = 0 \)),
- \( 0.959 \) is the decay factor.
The decay rate can be calculated as: \[ \text{Decay rate} = 1 - 0.959 = 0.041 \text{ or } 4.1% \]
Consumption in 2020
To find the consumption per person in the year 2020, we need to determine the value of \( t \) for that year. Since \( t \) represents the number of years since 2000:
\[ t = 2020 - 2000 = 20 \]
Now, we can substitute \( t = 20 \) into the consumption equation: \[ m(20) = 16.5(0.959)^{20} \]
Let's calculate \( (0.959)^{20} \): \[ (0.959)^{20} \approx 0.4096 \quad (\text{using a calculator}) \]
Now we can compute \( m(20) \): \[ m(20) = 16.5 \times 0.4096 \approx 6.7484 \]
Rounding to the nearest whole number gives us: \[ \text{Consumption per person in 2020} \approx 7 \text{ gallons} \]
Summary
- The rate of decay is approximately 4.1%.
- The approximate consumption of whole milk per person in 2020 is 7 gallons.