Write the equation of the model for the city’s population decay in terms of t
The Correct Answer and Explanation is:
The equation of a population decay model is typically written in the form:P(t)=P0e−ktP(t) = P_0 e^{-kt}P(t)=P0e−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 (which determines the rate of population decrease),
- ttt is the time that has passed.
Explanation:
The equation models exponential decay, which is a common way to represent populations that decrease over time (such as in the case of species extinction, urban depopulation, etc.).
- The initial population P0P_0P0 represents the population at time t=0t = 0t=0, when the decay process starts. This is the point at which the population is at its maximum before any decay has occurred.
- The decay constant kkk is a positive number that describes the rate at which the population decreases. A larger value of kkk means the population decreases more rapidly.
- The exponential function e−kte^{-kt}e−kt shows how the population declines over time. As ttt increases, the exponent becomes more negative, causing P(t)P(t)P(t) to decrease over time.
- The time variable ttt typically represents the amount of time that has passed since the start of the observation.
Example:
If the population of a city starts at 100,000 people and the population decreases at a rate of 2% per year, the decay constant kkk would be 0.020.020.02. The population model would then be:P(t)=100000e−0.02tP(t) = 100000 e^{-0.02t}P(t)=100000e−0.02t
This means that after 1 year, the population would be approximately P(1)=100000e−0.02×1≈98,000P(1) = 100000 e^{-0.02 \times 1} \approx 98,000P(1)=100000e−0.02×1≈98,000, showing a decrease of 2%. The exponential decay ensures the population will continue to decrease indefinitely, though the rate of decay slows down as time passes.
