The population in 20 years is 719 people.
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
According to the question
[tex]\frac{dP}{dt} = kP[/tex] where P is population at any time
solving equation,
[tex]\int[/tex][tex]\frac{dP}{dt} =\\[/tex] [tex]\int kP[/tex]
[tex]lnP=kt+C[/tex] -(1)
Applying the conditions,
at t=0, P = 500
[tex]ln500=C[/tex] -(2)
at t=10, P= 600
putting it in equation gives
[tex]ln(600) = k(10)+ln(500)[/tex]
ln(6/5) = k(10)
[tex]k = \frac{ln(6/5)}{10}[/tex]
at t=20 , P= ?
[tex]lnP=(\frac{ln(6/5)}{10})20 + ln500[/tex]
[tex]lnP=({ln(6/5)}{2 + ln500[/tex]
On putting the values of ln(6/5) , ln(500)
[tex]lnP=0.182(2) + 6.214[/tex]
[tex]P = e^{6.578}[/tex]
P = 719.099
On rounding off
P ≈ 719 people
Thus the population in 20 years is 719 people.
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