In 2000 the population of a state was 5 million and growing continuously at a rate of 3.4% per year.
a) Write a function that models the state’s population in millions x years after 200.
b) Graph f in 0,10,1 by 4,7,1
c) Estimate the state’s population in 2007.
To answer these questions, we need to use an exponential growth model. The formula for exponential growth is:
P(t) = P0 * e^(rt)
Where:
- P(t) is the population at time t
- P0 is the initial population
- e is the base of the natural logarithm (approximately 2.71828)
- r is the annual growth rate (expressed as a decimal)
- t is the time in years
a) To write a function that models the state's population in millions x years after 2000, we can plug in the given values:
P(t) = 5 * e^(0.034t)
You can use this function to calculate the population at any given year.
b) To graph this function in the given interval, we can create a table and plot the points. Let's find the population at each year from 0 to 10, with an increment of 1:
Year | Population (in millions)
---------------------------------
0 | 5
1 | ...
2 | ...
3 | ...
4 | ...
5 | ...
6 | ...
7 | ...
8 | ...
9 | ...
10 | ...
To find the population at each year, substitute the value of t in the function P(t).
c) To estimate the state's population in 2007, substitute t = 7 into the population function:
P(7) = 5 * e^(0.034 * 7)
Using a calculator or computer program, evaluate this expression to get the estimated population in 2007.