12. A city whose population was 39,350 in 1995 contained 46,750 people in 2000.
(a) Find the exponential growth function for the city. Let t = 0 represent the year 1995.
(b) Use the growth function to predict the population of the city in 2009. Round your answer to the nearest fifty people.
To find the exponential growth function for the city, we need to use the formula:
P(t) = P₀ * e^(kt)
Where:
P(t) = population at time t,
P₀ = initial population,
e = base of the natural logarithm,
k = growth rate,
t = time elapsed.
(a) We have two known points: in 1995, t=0 and P₀=39,350, and in 2000, t=5 years and P(5)=46,750. We can substitute these values into the equation to find k.
Using the point (0, 39,350):
P(0) = 39,350 = 39,350 * e^(0 * k)
1 = e^(0 * k)
Using the point (5, 46,750):
P(5) = 46,750 = 39,350 * e^(5 * k)
e^(5k) = 46,750 / 39,350 = 1.1879
Taking the natural logarithm (ln) of both sides:
5k = ln(1.1879)
k = ln(1.1879) / 5 ≈ 0.031
Therefore, the exponential growth function for the city is:
P(t) = 39,350 * e^(0.031t)
(b) To predict the population of the city in 2009, we need to substitute t = 14 (since 2009 - 1995 = 14) into the growth function:
P(14) = 39,350 * e^(0.031 * 14)
≈ 39,350 * e^(0.434)
≈ 39,350 * 1.544
≈ 60,762.2
Rounding to the nearest fifty people, the predicted population of the city in 2009 is 60,750.
To find the exponential growth function for the city, we can use the general form:
P(t) = P0 * e^(kt)
where P(t) is the population at time t, P0 is the initial population, k is the growth rate, and e is Euler's number, approximately equal to 2.71828.
Given that the population in 1995 was 39,350 (P0) and in 2000 was 46,750 (P(5)), we can substitute these values into the equation to find the value of k.
Using P0 = 39,350 and P(5) = 46,750, we have:
46,750 = 39,350 * e^(k*5)
Dividing both sides by 39,350, we get:
e^(5k) = 46,750 / 39,350
Taking the natural logarithm of both sides, we have:
ln(e^(5k)) = ln(46,750 / 39,350)
Using the property of logarithms, which says ln(e^x) = x, we can simplify it to:
5k = ln(46,750 / 39,350)
Now we can solve for k by dividing both sides by 5:
k = ln(46,750 / 39,350) / 5
Using a calculator or software, we find:
k ≈ 0.04355
Now that we have k, we can use it to find the exponential growth function. Plugging the values of P0 = 39,350 and k = 0.04355 into the equation:
P(t) = P0 * e^(kt)
We get:
P(t) = 39,350 * e^(0.04355t)
This is the exponential growth function for the city.
To predict the population of the city in 2009 using the growth function, we let t = 14 (since 2009 - 1995 = 14). Plugging this value into the equation:
P(14) = 39,350 * e^(0.04355 * 14)
Calculating this using a calculator or software, we find:
P(14) ≈ 39,350 * e^(0.6107) ≈ 39,350 * 1.8390 ≈ 72,251
Rounded to the nearest fifty people, the predicted population of the city in 2009 is 72,250.