The population P (in thousands) of Tallahassee, Florida, from 2000 through 2014 can be modeled by P = 150.9e^{kt}, where t represents the year, with t

abondantQ 2021-01-10 Answered
The population P (in thousands) of Tallahassee, Florida, from 2000 through 2014 can be modeled by P=150.9ekt, where t represents the year, with t=0 corresponding to 2000. In 2005, the population of Tallahassee was about 163,075.
(a) Find the value of k. Is the population increasing or decreasing? Explain.
(b) Use the model to predict the populations of Tallahassee in 2020 and 2025. Are the results reasonable? Explain.
(c) According to the model, during what year will the — populates reach 200,000?
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Expert Answer

Brittany Patton
Answered 2021-01-11 Author has 100 answers

Step 1
Given:
The population model is given as,
P=150.9ekt
a) To find the value of k, multiply ln on both sides.
P=150.9×1000ekt
163075=150.9×1000e5k
163075150900=e5k
ln×163075150900=ln(e5k)
ln×163075150900=5k
k=15(ln×163075150900)
k=15×0.0776
k=0.01552
k>0
Since the value of k is positive. Hence the population is increasing.
Step 2
b) The population in 2020 is,
t=20202000
t=20
P=150.9×1000×ekt
P=150900×e0.0155(20)
P=150900(1.3634)
P=205740
The population in 2025 is,
t=20252000
t=25
P=150.9×1000×ekt
P=150900×e0.0155(25)
P=150900×(1.4622)
P=220658
By comparing the population in 2005, the population in 2020 and 2025 is very low amount of growth.
Hence the obtained results are not reasonable.
Step 3
c) To find the year at which the population is 200000
P=150.9×1000×ekt
200000=150900×e0.0155t
200000150900=e0.0155t
Now taking ln on both sides,
ln×200000150900=ln(e0.0155t)
ln×200000150900=0.0155t
t=10.0155[ln×200000150900]
t=10.0155[ln1.32538]
t=10.0155[0.2817]
t=18.151
Hence in a time period of t 18 years the population will be 200000.

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