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Una población de bacterias crece de tal manera que la razón de crecimiento en el instante t (medido en horas) es igual a

p'(t) = 90t + 500 / 1 + t

Si el tamaño de la población inicial es de 2000 bacterias. Encontrar el tamaño de la población al cabo de 3 horas.

Sagot :

LammettHashidn

By the fundamental theorem of calculus, if

p'(t) = 90t + 500/(1 + t)

and p(0) = 2000, then

p(t) = p(0) + ∫₀ᵗ p'(u) du

⇒   p(t) = 2000 + ∫₀ᵗ (90u + 500/(1 + u)) du

⇒   p(t) = 2000 + (45u² + 500 ln|1 + u|)₀ᵗ

⇒   p(t) = 2000 + 45t² + 500 ln|1 + t|

Then after t = 3 hours, the population of bacteria would be

p(3) = 2000 + 45•3² + 500 ln(1 + 3) ≈ 3098

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