Question

Let P (t) be the population of the United States (in millions), where t is the number of years after 1900.

The population of the US in 1900 was 76 million, so Po = P (0) = 76. The population was 123 million in
1930, 151 million in 1950, and 249 million in 1990 . Use this information, and assuming logistic growth, to
find the predicted population in 2010. Compare the predicted value to the actual 2010 population of 309
million.

Answer 1

To find the predicted population in 2010 using logistic growth, use the logistic growth equation and the given information. Solve for the carrying capacity (K) using the known population in 1990. Then, plug in the values to find the predicted population in 2010 and compare it to the actual population.

Step-by-step explanation:

To find the predicted population in 2010 using logistic growth, we can use the logistic growth equation, which is given by:

P(t) = (K / (1 + (K/P0 - 1) * e^(-r(t-t0))))

Here, P(t) is the population at time t, K is the carrying capacity, P0 is the initial population, r is the growth rate, t0 is the initial time, and e is Euler's number.

Using the given information, we can calculate the parameters:

Initial population (P0) = 76 million

Population in 1990 (P(90)) = 249 million

Time in years (t) = 1990 - 1900 = 90

Plugging these values into the equation, we can solve for the carrying capacity (K):

249 = (K / (1 + (K/76 - 1) * e^(-r(90))))

By solving this equation, we can find the value of K.

Once we have the value of K, we can use it to find the predicted population in 2010 by plugging in the values of P0, t, and t0 into the logistic growth equation:

P(2010) = (K / (1 + (K/P0 - 1) * e^(-r(2010-t0))))

Compare the predicted population to the actual population to see the difference.