Jim's family recently moved to a new city. The city's population has been growing, and based on recent trends, Jim expects it to continue growing exponentially. This table shows the expected population in the next two years.
Time (years) Population
1 840,000
2 882,000
Find an expression for P(t), the city's population t years from now. Write your answer in the form P(t)=a(b)t, where a and b are integers or decimals. Do not round.
P(t)=
To find an expression for P(t), we can observe that the city's population is growing exponentially. This means that each year the population is increasing by a certain percentage.
From the given data, we can see that the population has increased by 42,000 (882,000 - 840,000) from year 1 to year 2.
Therefore, we can say that the population grows by 42,000/840,000 = 0.05 or 5% per year.
Now, let's find the initial population, P(0), which is the population at time t = 0. From the given data, we can observe that the population at time t = 1 (1 year from now) is 840,000.
Using the formula for exponential growth, we can express P(t) as:
P(t) = P(0)*(1 + r)^t
where P(0) is the initial population, r is the annual growth rate as a decimal, and t is the number of years from now.
Now, substitute the values into the formula:
P(t) = 840,000*(1 + 0.05)^t
Therefore, the expression for P(t), the city's population t years from now, is:
P(t) = 840,000*(1.05)^t