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

Kyran Hudson 2021-02-25 Answered
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 t years after 1990.
(b) Find the time required for the population to double.
(c) Use the function from part (a) to predict the population of California in the year 2010. Look up California’s actual population in 2010, and compare.
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Expert Answer

Sally Cresswell
Answered 2021-02-26 Author has 91 answers

The population of California was 29.76 million in 1990 and 33.87 million in 2000. Assume that the population grows exponentially.
Let A0 and r be the initial population & growth rate of population respectively. Then population in California t years after 1990 is,
P(t)=A0ert (1)
a. Since the population in 1990 is 29.76 million, therefore A0=29.76
In 2000, the population is 33.87 million. That is,
P(t)=33.87 when t=10
Therefore from equation (1), we get
33.87=29.76e10r
e10r=1.13810483
10r=ln(1.13810483)
r=ln(1.13810483)10
r=0.012936
Therefore the population t years after 1990 is,
P(t)=29.76e0.012936t
b) The double of 29.76 million is 59.52 million.
So put P(t)=59.52 in P(t)=29.76e0.012936t, we get
59.52=29.76e0.012936t
e0.012936t=2
0.012936t=ln(2)
t=ln(2)0.012936
t=53.5828
Hence the population will double in 53.5828 years.
c)Since 2010−1990=20, therefore to predict the population in 2010, put t=20 in P(t)=29.76e0.012936t , we get
P(20)=29.76e0.012936(20)
=38.5472 million
The population in California in 2010 is 37.3 million, therefore the actual population is approximately 1 million less than the obtained population.

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