Answer:
The maximum concentration of the antibiotic during the first 12 hours will be of 0.5926 µg/ml.
Explanation:
The maximum concentration will be at the instant of time [tex]t = t'[/tex] in which [tex]c'(t) = 0[/tex], and this concentration will be [tex]c(t')[/tex].
So
[tex]c(t) = 4(e^{-0.4t} - e^{-0.6t})[/tex]
[tex]c'(t) = -1.6e^{-0.4t} + 2.4e^{-0.6t}[/tex]
[tex]c'(t) = 0[/tex]
[tex]-1.6e^{-0.4t} + 2.4e^{-0.6t} = 0[/tex]
[tex]2.4e^{-0.6t} = 1.6e^{-0.4t}[/tex]
[tex]e^{-0.2t} = \frac{1.6}{2.4}[/tex]
Applying ln to both sides
[tex]\ln{e^{-0.2t}} = \ln{\frac{1.6}{2.4}}[/tex]
[tex]-0.2t = -0.4054[/tex]
[tex]t = 2.027[/tex]
The maximum concentration will happens at the time 2.027h, and it will be
[tex]c(2.027) = 4(e^{-0.4*2.027} - e^{-0.6*2.027}) = 0.5926[/tex]
The maximum concentration of the antibiotic during the first 12 hours will be of 0.5926 µg/ml.
The concentration of the antibiotic after 12 hours is 0.0299 µg/ml.
Antibiotics are drugs that are taken to combat diverse bacterial infections. There are several antibiotics on sale and they are used for diseases such as typhoid fever, cholera, pneumonia, etc.
Following the model of the function, we have;
c(t) = 4(e−0.4t − e−0.6t), the concentration of the drug is a function of time.
At time = 12 hours, we have;
c(t) = 4(e−0.4(12) − e−0.6(12))
c(t) = 4(8.2297 × 10^-3 - 0.7466 × 10^-3)
c(t) = 0.0299 µg/ml
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