Question

The number of bacteria, measured in thousands, in a culture is modeled by the equation $b\left(t\right)=\frac{380e^{2.3lt}}{175+e^{3.2lt}},$ where $t$ is the number of days since the culture was formed. According to this model, the culture can support a maximum population of

• A. $217$
• B. $205$
• C. $380$
• D. $760$
• E. $\infty$

Answer 1

Answer: (C)

$380$

Given $b\left(t\right)=380\left(\frac{e^{2.31t}}{175+e^{3.21t}}\right)$

$b^{\prime }\left(t\right)=380\left(\frac{e^{2.31t}(175+e^{3.21t})-3.21\times e^{3.21t}\times e^{2.31t}}{175+e^{3.21t}}\right)$

$0=2.31\times 175+2.31e^{3.21t}-3.21e^{3.21t}$

$e^{3.21t}=\frac{2.31\times 175}{3.21-2.31}=449.177$

$3.21t=6.107$

$t=1.9$

$b\left(t\right)=380$