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The population of a town grows at a rate proportional to the population present at time t. The initial population of 500 increases by 10% in 10 years. What will be the population in 20 years? (Round your answer to the nearest person.)
510

Incorrect: Your answer is incorrect.
persons
How fast (in persons/yr) is the population growing at
t = 20?
(Round your answer to two decimal places.)
persons/yr

Sagot :

Cricetus

Answer:

The right answer is "605 persons".

Step-by-step explanation:

Given that,

⇒ [tex]\frac{dP}{dt} \alpha P[/tex]

or,

⇒ [tex]\frac{dP}{dt} =RP[/tex]

⇒ [tex]\frac{1}{R} \int\limits^{PT}_{500} {\frac{dP}{P} } = \int\limits^{t}_{0} dt[/tex]

⇒ [tex]\frac{1}{R} [log PT-log500]=t[/tex]

⇒                  [tex]log[\frac{PT}{500} ]=Rt[/tex]

At t = 10 years,

⇒ [tex]PT=\frac{110}{100}\times 500[/tex]

          [tex]=550[/tex]

∴ [tex]log[\frac{550}{500} ]=R\times 10[/tex]

            [tex]R=\frac{log(1.1)}{10}[/tex]

At t = 20,

⇒ [tex]log{\frac{PT}{500} }=\frac{log \ 1.1}{10}\times 20[/tex]

        [tex]PT=500\times (1.1)^2[/tex]

              [tex]=605 \ persons[/tex]

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