The value of [tex]k[/tex] is approximately 14.426.
How to analyze a first order differential equation
According to statement, population grows exponentially based on the fact that it is doubled every 10 years. The solution of this differential equation is:
[tex]t = k\cdot \ln \frac{y}{y_{o}}[/tex] (1)
Where:
- [tex]k[/tex] - Proportionality ratio, in [tex]\frac{1}{yr}[/tex].
- [tex]y_{o}[/tex] - Initial population
- [tex]y[/tex] - Current population
- [tex]t[/tex] - Time, in years.
If we know that [tex]t = 10\,yr[/tex] and [tex]y = 2\cdot y_{o}[/tex], then the proportionality ratio is:
[tex]10 = k\cdot \ln 2[/tex]
[tex]k = \frac{10}{\ln 2}[/tex]
[tex]k \approx 14.426[/tex]
The value of [tex]k[/tex] is approximately 14.426. [tex]\blacksquare[/tex]
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