A bacterial infection in an internal wound grows as N ' t = N 0 e x p ⁡ ( t ) , where the time t is in hours. A dose of antibiotic, taken orally, needs 1 hour to reach the wound. Once it reaches there

Question

A bacterial infection in an internal wound grows as N ' t = N 0 e x p ⁡ ( t ) , where the time t is in hours. A dose of antibiotic, taken orally, needs 1 hour to reach the wound. Once it reaches there, the bacterial population goes down as d N d t = - 5 N 2 . What will be the plot of N 0 N vs t after 1 hour?

TestHub · Accepted Answer

From 0 to 1 hour, N ' = N 0 e t From 1 hour onwards d N d t = - 5 N 2 So at t = 1 hour, N ' = e N 0 d N d t = - 5 N 2 ∫ e N 0 N N - 2 d N = - 5 ∫ 1 t d t 1 N - 1 e N 0 = 5 t - 1 N 0 N - 1 e = 5 N 0 ( t - 1 ) N 0 N = 5 N 0 ( t - 1 ) + 1 e N 0 N = 5 N 0 t + 1 e - 5 N 0 Which is following y = m x + C

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