The population of the US in 1850 was 23191876. In 1900, the population was 62947714.
a. assuming that the population grew exponentially, compute the grownth constant.
b. assuming continued growth at a constant rate, predict the population in 1950.
let t=0 correspond with the year 1850
then 1900 ---> t = 50
then 1950 ---> t = 100
population = a( e^(kt) ) , where a is the initial population and t is the number of years since 1850
23,191,876 = a e^0
a = 23,191,876
then:
62,947,714 = 23,181,876 e^(50k)
2.715384812 = e^(50k)
50k = ln(2.715384812 )
k = .019978673
in 1950
pop = 23,191,876 e^(100( .019978673))
= 170,927,267
a. To determine the growth constant assuming exponential growth, we can use the formula:
P(t) = P0 * e^(k*t)
where P(t) is the population at time t, P0 is the initial population, e is Euler's number (approximately 2.71828), k is the growth constant, and t is the time in years.
Given the population in 1850 (P0) as 23191876 and the population in 1900 (P(50)) as 62947714, we can substitute these values into the formula:
62947714 = 23191876 * e^(k*50)
To find the growth constant (k), we need to isolate it in the equation. First, divide both sides of the equation by 23191876:
(62947714 / 23191876) = e^(k*50)
Next, take the natural logarithm (ln) of both sides to remove the exponential function:
ln(62947714 / 23191876) = k*50
Now, divide the left side by 50 to solve for k:
k = ln(62947714 / 23191876) / 50
Using a calculator, compute the natural logarithm of the ratio and divide by 50 to find the growth constant.
b. To predict the population in 1950 assuming continued growth at a constant rate, we can use the formula:
P(t) = P0 + (r * t)
where P(t) is the population at time t, P0 is the initial population, r is the growth rate, and t is the time in years.
Given the population in 1900 (P0) as 62947714 and the year 1950 (t) as 1950-1900 = 50, we can substitute these values into the formula:
P(50) = 62947714 + (r * 50)
To find the growth rate (r), we need to isolate it in the equation. First, subtract the initial population from both sides of the equation:
P(50) - 62947714 = r * 50
Now, divide both sides of the equation by 50 to solve for r:
r = (P(50) - 62947714) / 50
Substitute the value of P(50) to get the predicted population in 1950.