Question

The population of California was 29.76 million in 1990 and 33.87 million in 2000. Assume that the population grows exponentially.

(a) Find a function that models the population (in millions) t years after 1990. (Round your r value to six decimal places.)
(b) Find the time required after 1990 for the population to double. (Round your answer to one decimal place.)
(c) Use the function from part (a) to predict the population of California in the year 2006. (Round your answer to two decimal places.)

Answer 1

The exponential growth function of California's population is modeled as P(t) = 29.76e^{0.013028t}, where t is the number of years after 1990. The time for the population to double is approximately 53.2 years, and the predicted population in 2006 is approximately 37.57 million people.

Step-by-step explanation:

The population of California in 1990 was 29.76 million and in 2000 it was 33.87 million. When assuming exponential growth, we can model this with the function:

P(t) = P0ert

Where P(t) is the population t years after 1990, P0 is the initial population, e is the base of natural logarithms, and r is the growth rate.

To find r, we use the given data points for 1990 (t=0) and 2000 (t=10):

33.87 = 29.76er(10)

Solving for r gives us:

r ≈ 0.013028 (rounded to six decimal places)

The resulting function is:

P(t) = 29.76e0.013028t

To find the time required for the population to double, we set P(t) equal to twice the initial population:

59.52 = 29.76e0.013028t

Solving for t, we get:

t ≈ 53.2 years (rounded to one decimal place).

For the prediction of population in 2006: t = 16 (since 2006 is 16 years after 1990),

P(16) ≈ 37.57 million (rounded to two decimal places)

Answer 2

To model the population of California with exponential growth, the formula P(t) = 29.76e^(0.013046t) is used, with t being years after 1990. It would take approximately 53.3 years for the population to double. The estimated population for 2006 using this model is about 35.755 million.

Step-by-step explanation:

Population Growth Modeling

Given the exponential growth model, we can define a population function P(t) that models the population in millions t years after 1990 with P(0) = 29.76 and P(10) = 33.87. We can use the formula for exponential growth, P(t) = P0 * e^(rt), where P0 is the initial population, r is the growth rate, and t is the time in years

Part (a): Finding the Population Function

We have two points, (0, 29.76) and (10, 33.87), which can be used to find r:

33.87 = 29.76 * e^(10r) ⇐ r ≈ 0.013046 (rounded to six decimal places)

The population function is therefore P(t) = 29.76e^(0.013046t).

Part (b): Time to Double the Population

The time T needed to double can be found using P(T) = 2 * P0:

59.52 = 29.76 * e^(0.013046T) ⇐ T ≈ 53.3 years (rounded to one decimal place)

Part (c): Predicting Population in 2006

To predict the population in 2006, we use t = 16:

P(16) = 29.76 * e^(0.013046*16) ≈ 35.755 million (rounded to two decimal places)