A given field mouse population satisfies the differential equation dp/dt=.4p-450 where p is the number of mice and t is the time in months.
(a) Find the time at which the population becomes extinct if p(0)=1075
Round your answer to two decimal places in months.
(b) Find the initial population Po if the population is to become extinct in 1 year.
Round your answer to the nearest integer.
Po= mice

Answer:

a) 7.78 months

b) 1116

Step-by-step explanation:

Given -

[tex]\frac{dP}{dt} = 0.4p-450[/tex]

Integrating the above equation with respect to time, we get -

[tex]\int\ \frac{dP}{0.4p-450} = \int\ dt\\[/tex]

let us define new variable x

[tex]x = 0.4p - 450 \\dx = 0.4 dy[/tex]

substituting these values in above integral equation, we get -

[tex]\frac{1}{0.4} \int\ \frac{dx}{x} = \int\ dt\\ln x = 0.4 t + C\\x = ce^{0.4t}[/tex]

[tex]P (t) = \frac{C}{0.4} e^{0.4t} +1125\\[/tex]

at t = 0, P (0) = 1075, using this condition, we get -

[tex]\frac{c}{0.4} = -50\\P(t) = 50 e^{0.4t} +1125 \\t = 7.78\\P(0) = 1125 - \frac{1125}{e^{4.8}} = 1116[/tex]

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