Step-by-step explanation:
using DE exponential growth formula, the equation will be,
[tex] \frac{dp}{dt} = gen[/tex]
where
dP/dt = number of population per unit time
gen = generated population
[tex] \frac{dp}{dt} = 0.4p[/tex]
gen is always a function of P
[tex] \frac{dp}{dt} \times dt = (0.4p)dt \\ \\ dp = (0.4p)dt[/tex]
use variable seperable technique then integrate
[tex] \frac{dp}{p} = 0.4 dt \\ \\ ln( |p| ) = 0.4(t + c)[/tex]
raise all sides of the equation to become the power of e
[tex] {e}^{ ln( |p| ) } = {e}^{0.4t + 0.4c} [/tex]
by the rules of exponential, e raise to ln will cancel out leaving only P
[tex] {e}^{0.4t + 0.4c} [/tex]
by the rules of exponents, this can be written as,
[tex] {e}^{0.4t} \: and \: {e}^{0.4c} [/tex]
0.4C is just a constant so it will just become e^C and e^C is just also a constant and thus just becomes C
general solution
[tex]p = c {e}^{0.4t} [/tex]
remember that initially(t=0) the population is at 25000(P=25k), substituting to the equation we get,
[tex]25000 = c {e}^{0.4(0)} \\ 25000 = c {e}^{0} \\ 25000 = c[/tex]
particular solution
[tex]p = 25000 {e}^{0.4t} [/tex]
to get the population at t=8hours, substitute 8 to t
[tex]p = 25000 {e}^{0.4(8)} \\ p = 613313.25[/tex]
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Answers & Comments
Step-by-step explanation:
using DE exponential growth formula, the equation will be,
[tex] \frac{dp}{dt} = gen[/tex]
where
dP/dt = number of population per unit time
gen = generated population
[tex] \frac{dp}{dt} = 0.4p[/tex]
gen is always a function of P
[tex] \frac{dp}{dt} \times dt = (0.4p)dt \\ \\ dp = (0.4p)dt[/tex]
use variable seperable technique then integrate
[tex] \frac{dp}{p} = 0.4 dt \\ \\ ln( |p| ) = 0.4(t + c)[/tex]
raise all sides of the equation to become the power of e
[tex] {e}^{ ln( |p| ) } = {e}^{0.4t + 0.4c} [/tex]
by the rules of exponential, e raise to ln will cancel out leaving only P
[tex] {e}^{0.4t + 0.4c} [/tex]
by the rules of exponents, this can be written as,
[tex] {e}^{0.4t} \: and \: {e}^{0.4c} [/tex]
0.4C is just a constant so it will just become e^C and e^C is just also a constant and thus just becomes C
general solution
[tex]p = c {e}^{0.4t} [/tex]
remember that initially(t=0) the population is at 25000(P=25k), substituting to the equation we get,
[tex]25000 = c {e}^{0.4(0)} \\ 25000 = c {e}^{0} \\ 25000 = c[/tex]
particular solution
[tex]p = 25000 {e}^{0.4t} [/tex]
to get the population at t=8hours, substitute 8 to t
[tex]p = 25000 {e}^{0.4(8)} \\ p = 613313.25[/tex]