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NO LINKS! PLEASE HELP ME!​

NO LINKS! PLEASE HELP ME!​-example-1
User Chkdsk
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1 Answer

5 votes

Answer:

4860 bacteria

Explanation:

The given scenario can be modelled using an exponential function with base e.


\boxed{\begin{minipage}{9 cm}\underline{General form of an Exponential Function with base $e$}\\\\$y=Ae^(kx)$\\\\where:\\\phantom{ww}$\bullet$ $A$ is the initial value ($y$-intercept). \\ \phantom{ww}$\bullet$ $e$ is Euler's number. \\ \phantom{ww}$\bullet$ $k$ is some constant.\\\end{minipage}}

Define the variables:

  • Let y be number of bacteria.
  • Let x be the time (in hours)

Given the initial number of bacteria is 20, then A = 20.


\implies y=20e^(kx)

If the bacteria triples its number in 16 hours, then when x = 16:


\implies y=3 * 20= 60

Substitute x = 16 and y = 60 into the equation and solve for k:


\begin{aligned}y&=20e^(kx)\\\implies 60&=20e^(16k)\\(60)/(20)&=e^(16k)\\3&=e^(16k)\\\ln 3&=\ln e^(16k)\\ \ln3&=16k \ln e\\\ln3&=16k\\k&=(1)/(16) \ln 3\end{aligned}

Therefore, the equation that models the given scenario is:


\large\boxed{y=20e^{\left((1)/(16)x \ln 3\right)}}

To find how many bacteria there are after 80 hours, substitute x = 80 into the equation and solve for y:


\implies y=20e^{\left((1)/(16)(80) \ln 3\right)}


\implies y=20e^(\left(5 \ln 3\right))


\implies y=20e^(\left(\ln 3^5\right))


\implies y=20e^(\left(\ln 243\right))


\implies y=20 \cdot 243


\implies y=4860

Therefore, there are 4860 bacteria after 80 hours.

-----------------------------------------------------------------------------------------------

Please note that I have solved this question so that you can calculate the number of bacteria for any number of hours, part hours, etc.

However, there is a much simpler way of solving this particular question.

If the number of bacteria triples every 16 hours, then it will triple 5 times within 80 hours since 80 ÷ 16 = 5.

Therefore:

  • Hour 0: 20
  • Hour 16: 20 × 3 = 60
  • Hour 32: 60 × 3 = 180
  • Hour 48: 180 × 3 = 540
  • Hour 64: 540 × 3 = 1620
  • Hour 80: 1620 × 3 = 4860

As you can see, the result is the same as the previous method.

User Virtua Creative
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