Question

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The number of bacteria in a culture is increasing according to the law of exponential growth. The initial population is 240 bacteria, and the population after 7 hours is double the population after 1 hour. How many bacteria will be there after 4 hours? (Round your answers to the nearest whole number.)

Answer 1

Answer:

381 bacteria

Step-by-step explanation:

$\boxed{\begin{minipage}{9 cm}\underline{General form of an Exponential Function with base e }\\\\ y=Ae^{kt} \\\\where:\\\phantom{ww} \bullet A is the initial value ( y -intercept). \\ \phantom{ww} \bullet e is Euler's number. \\ \phantom{ww} \bullet k is some constant.\\ \phantom{ww} \bullet t is the independent value time.\\\end{minipage}}$

If the initial population is 240 bacteria, then A = 240:

$\implies y=240e^{kt}$

Create equations for the population after 1 hour and 7 hours:

$t=1\implies y=240e^{k}$

$t=7\implies y=240e^{7k}$

Given the population after 7 hours is double the population after 1 hour:

$\implies 240e^{7k}=2 \cdot 240e^k$

$\implies e^{7k}=2e^k$

Solve the equation to find the value of k:

$\implies e^{7k}=2e^k$

$\implies \dfrac{e^{7k}}{e^k}=2$

$\implies e^{6k}=2$

$\implies \ln e^{6k}=\ln 2$

$\implies 6k \ln e = \ln 2$

$\implies 6k=\ln 2$

$\implies k=\dfrac{1}{6}\ln 2$

Therefore, the equation that models the given parameters is:

$y=240e^{\frac{1}{6}t\ln 2}$

To find how many bacteria there will be after 4 hours, substitute t = 4 into the equation:

$\implies y=240e^{\frac{1}{6}(4)\ln 2}$

$\implies y=240e^{\frac{2}{3}\ln 2}$

$\implies y=240(1.58740105...)$

$\implies y=380.9762525$

$\implies y=381$

Therefore, there will be 381 bacteria after 4 hours