Answer:
The maximum concentration of the antibiotic during the first 12 hours is 1.037 µg/mL
Step-by-step explanation:
The maximum concentration of the antibiotic during the first 12 hours will happen when:
[tex]C'(t) = 0[/tex]
We have that:
[tex]C(t) = 7(e^{-0.4t} - e^{-0.6t})[/tex]
[tex]C(t) = 7e^{-0.4t} - 7e^{-0.6t}[/tex]
The derivative of [tex]e^{at}[/tex] is [tex]ae^{at}[/tex] and the derivative of a sum/subtraction is the sum/subtraction of the derivatives.
So
[tex]C'(t) = -2.8e^{-0.4t} + 4.2e^{-0.6t}[/tex]
[tex]C'(t) = 0[/tex]
[tex]-2.8e^{-0.4t} + 4.2e^{-0.6t} = 0[/tex]
[tex]4.2e^{-0.6t} = 2.8e^{-0.4t}[/tex]
[tex]\frac{e^{-0.6t}}{e^{-0.4t}} = \frac{2.8}{4.2}[/tex]
[tex]e^{-0.2t} = 0.6667[/tex]
[tex]\ln{e^{-0.2t}} = \ln{0.0667}[/tex]
[tex]-0.2t = -0.4055[/tex]
[tex]0.2t = 0.4055[/tex]
[tex]t = \frac{0.4055}{0.2}[/tex]
[tex]t = 2.03[/tex]
The maximum concentration is
C(2.03)
[tex]C(t) = 7e^{-0.4t} - 7e^{-0.6t}[/tex]
[tex]C(2.03) = 7e^{-0.4*2.03} - 7e^{-0.6*2.03} = 1.037[/tex]
The maximum concentration of the antibiotic during the first 12 hours is 1.037 µg/mL
