Write the equation of the model for the city’s population decay in terms of t
The Correct Answer and Explanation is:
To model the population decay of a city over time, we can use the general exponential decay equation:P(t)=P0⋅e−ktP(t) = P_0 \cdot e^{-kt}P(t)=P0⋅e−kt
Where:
- P(t)P(t)P(t) is the population at time ttt,
- P0P_0P0 is the initial population at time t=0t = 0t=0,
- kkk is the decay constant (a positive number for decay),
- ttt is the time variable, typically in years, months, or another time unit.
Explanation:
In this equation, the population decreases exponentially over time. The factor e−kte^{-kt}e−kt represents the rate of decay. Exponential decay is characterized by a constant percentage reduction over each time interval. This means that the longer the time passes, the smaller the population becomes, following a consistent pattern.
- Initial Population: The value P0P_0P0 represents the initial population of the city at the beginning of the time period. It is the population at t=0t = 0t=0, and it determines the starting value for the model.
- Decay Constant: The decay constant kkk determines how quickly the population decreases. The larger the value of kkk, the faster the decay. For example, if k=0.05k = 0.05k=0.05, the population decays at a rate of 5% per unit of time. A smaller kkk means a slower rate of decay.
- Time Variable: The time variable ttt represents the time that has passed since the initial population measurement. As ttt increases, the population P(t)P(t)P(t) decreases exponentially.
This model assumes that the city’s population decreases without external factors like immigration or significant changes in birth rates. It reflects situations where a population declines at a steady, predictable rate.
