In 1980, the population of a certain country was about 161 000. since then the population has decreased about 1% per year.
a. find an equation to model the population since 1980
b. estimate the population in 1990
c. suppose the current trend continues. predict the population 2000.
P(t) = 161000(.99)^t, where t is the number of years since 1980
b) evaluate for t = 10
c) evaluate for t = 20
(time to update your textbook)
To find the equation that models the population since 1980, we need to consider that the population has been decreasing about 1% per year. We can start with the initial population in 1980 and then apply the decrease rate for each subsequent year.
a. Finding the equation to model the population since 1980:
Let's define the population in 1980 as P₀ = 161,000.
Since the population decreases 1% each year, we can write the equation as:
P(t) = P₀ * (1 - r)^t
Where:
- P(t) represents the population at time t.
- P₀ is the initial population in 1980, which is 161,000.
- r is the rate of decrease, which is 1% or 0.01.
- t is the number of years since 1980.
b. Estimating the population in 1990:
To estimate the population in 1990, we need to substitute t = 10 (since 1990 is 10 years after 1980) into the equation we derived above:
P(10) = 161,000 * (1 - 0.01)^10
P(10) ≈ 161,000 * 0.9^10
P(10) ≈ 161,000 * 0.3487
P(10) ≈ 56,110.7
The estimated population in 1990 is approximately 56,111.
c. Predicting the population in 2000:
Similarly, to predict the population in 2000, we substitute t = 20 (since 2000 is 20 years after 1980) into the equation:
P(20) = 161,000 * (1 - 0.01)^20
P(20) ≈ 161,000 * 0.9^20
P(20) ≈ 161,000 * 0.1216
P(20) ≈ 19,581.6
The predicted population in 2000 is approximately 19,582.
Please note that these figures are estimates based on the given assumptions, and may not reflect the actual population values. They are intended to demonstrate the calculation method.