$\"\"$

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(a)

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The amount $\"\"$ is the number of bacteria in the culture and $\"\"$ is the time in hours,then $\"\"$ as a function of $\"\"$ is $\"\"$ .

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Where $\"\"$ is the initial population of the colony and $\"\"$ is a rate of decay.

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Therefore,the function is $\"\"$ .

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$\"\"$

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(b)

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The function is $\"\"$ .

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The initial bacteria in the culture is $\"\"$ .

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The population after one hour is $\"\"$ .

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The time period is $\"\"$ .

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Substitute all the values in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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Take natural logarith on each side.

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$\"\"$

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$\"\"$

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The rate of decay is $\"\"$ per hour.

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$\"\"$

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The function is $\"\"$ .

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Find the popluation after five hours,then $\"\"$ .

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Substitute $\"\"$ and $\"\"$ in the function.

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$\"\"$

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$\"\"$

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The population of bacteria in the culture after five years is $\"\"$ .

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$\"\"$

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(c)

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Find the number of hours when the population of the bacteria in the culture is $\"\"$ .

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The function is $\"\"$ .

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Substitute $\"\"$ and $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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Take natural logarithm on each side.

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$\"\"$

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$\"\"$

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$\"\"$

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The population of the bacteria in the culture is $\"\"$ after $\"\"$ hours.

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$\"\"$

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(a) The function is $\"\"$ .

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(b) The population of bacteria in the culture after five years is $\"\"$ .

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(c) The population of the bacteria in the culture is $\"\"$ after $\"\"$ hours.