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]
