According to the U.S Bureau of Census in the year 1850 the population of the US was 23,191,876 in 1900 the population was 62,947,714
A. Assuming that the population grew exponentially during the period compute
the growth constant k.
B. Assuming continued growth at the same rate predict the 1950 population
p =pi e^kt
let t = 0 at 1850
so pi = 23,191,876
p = 23,191,876 e^kt
in 1900 t = 1900-1850 = 50
so
62,947,714 = 23,191,876 e^50k
1.275 = e^50k
take ln of both sides
50 k = .2429
k = .00486
P at t is 100 years = 23,191,876 e^(.486)
= 23,191,876 (1.6258)
=37,705,351
I approximated all the natural logs so you should do the arithmetic more accurately although obviously the census and law are not that accurate anyway.
p =pi e^kt
let t = 0 at 1850
so pi = 23,191,876
p = 23,191,876 e^kt
in 1900 t = 1900-1850 = 50
so
62,947,714 = 23,191,876 e^50k
2.714 = e^50k
take ln of both sides
50 k = .9984
k = .01997
P at t is 100 years = 23,191,876 e^(1.997)
= 23,191,876 (7.367)
=170,852,745
I approximated all the natural logs so you should do the arithmetic more accurately although obviously the census and law are not that accurate anyway.
THank you so much
To find the growth constant (k) assuming exponential growth, we can use the formula:
P = P0 * e^(kt)
Where:
P = Final population
P0 = Initial population
e = Euler's number (approximately 2.718)
k = Growth constant
t = Time period
We have two data points:
In 1850, P0 was 23,191,876 and in 1900, P was 62,947,714.
A. To find the growth constant (k), we need to solve for it using the given data points.
Using the formula, we can set up two equations:
P0 = P0 * e^(k * 0)
62,947,714 = 23,191,876 * e^(k * 50)
Simplifying the equations:
1 = e^(k * 0)
e^(50k) = 62,947,714 / 23,191,876
Now, take the natural logarithm of both sides to solve for k:
ln(e^(50k)) = ln(62,947,714 / 23,191,876)
50k * ln(e) = ln(62,947,714 / 23,191,876)
50k = ln(62,947,714 / 23,191,876)
Finally, solve for k:
k = ln(62,947,714 / 23,191,876) / 50
B. Now that we have the growth constant (k),
we can use the exponential growth formula to predict the 1950 population.
Let's solve for P (population in 1950) using the formula:
P = P0 * e^(kt)
P0 = 62,947,714 (population in 1900)
t = 1950 - 1900 = 50 years (time period)
P = 62,947,714 * e^(k * 50)
Simply plug in the value of k that we found in part A, and calculate P to get the predicted population in 1950.