Population Growth. Port St. Lucie, Florida, had the United States fastest growth rate among cities with a population of 100,000 or more between 2003 and 2004. In 2003, the population was 103, 800 and increasing at a rate of 12% per year. In what year should the population reach 209,000 ? (Let t = 0 correspond to 2003.) Apply the formula N = N0 e^rt, where N represents the number of people. Round your answer to one desimal place. T = ______________years.The tolerance is ±2%

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slicergiza slicergiza · Answer 1 · 2019-12-11T10:09:10+01:00

Answer:

In 2009

Step-by-step explanation:

Since, the formula of population after t years,

[tex]N=N_0 e^{rt}[/tex]

Where,

r = rate of growing per year,

Here, r = 12% = 0.12,

So, the population formula would be,

[tex]N=N_0 e^{0.12t}[/tex]

If the population is estimated since 2003,

i.e. for 2003, t = 0,

We have N = 103, 800 for 2003,

[tex]\implies 103800 = N_0 e^{0.12\times 0}=N_0 e^0 = N_0[/tex]

Thus, the function that shows the population after t years,

[tex]N = 103800 e^{0.12t}[/tex]

If N = 209,000,

[tex]209000 = 103800 e^{0.12t}[/tex]

[tex]\frac{209000}{103800} = e^{0.12t}[/tex]

[tex]2.01349 = e^{0.12t}[/tex]

Taking ln on both sides,

[tex]\ln ( 2.01348 ) = 0.12t[/tex]

[tex]\implies t = \frac{\ln(2.01348)}{0.12}=5.8\approx 6[/tex]

∵ 2003 + 6 = 2009

Hence, in 2009, the population should reach 209,000.