# A water sample in a laboratory initially contains 6000 bacteria. The organisms reproduce at a rate of 10% per hour. Find the function that corresponds to this situation. Then predict how long it will take for the population of bacteria to double in number. Round your answer to the nearest hundredth of an hour.

Question
Exponential growth and decay
A water sample in a laboratory initially contains 6000 bacteria. The organisms reproduce at a rate of 10% per hour. Find the function that corresponds to this situation. Then predict how long it will take for the population of bacteria to double in number. Round your answer to the nearest hundredth of an hour.

2020-10-22
Use the exponential growth function:
y=a(1+r)x
where aa is the initial value and r is the growth rate.
Substitute a=6000 (6000 bacteria) and r=0.1 (from 10% per hour) to find the equation:
$$\displaystyle{y}={6000}{\left({1}+{0.1}\right)}^{{x}}$$
$$\displaystyle{y}={6000}{\left({1.1}\right)}^{{x}}$$
Substitute y=12000 (double in number) and solve for x:
$$\displaystyle{12000}={6000}{\left({1.1}\right)}^{{x}}$$
Divide both sides by 6000:
$$\displaystyle{2}={\left({1.1}\right)}^{{x}}$$
Take the natural logarithm of both sides:
$$\displaystyle{\ln{{2}}}={{\ln{{\left({1.1}\right)}}}^{{x}}}$$
Apply Power Property of logarithms:
$$\displaystyle{\ln{{2}}}={x}{\ln{{1.1}}}$$
Divide both sides by ln1.1:
$$\displaystyle\frac{{\ln{{2}}}}{{\ln{{1.1}}}}={x}$$
x≈7.27 hours

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