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After a special medicine is introduced into a petri dish containing a bacterial culture, the number of bacteria remaining in the dish decreases rapidly.
The population loses
(1)/(4) of its size every 44 seconds. The number of remaining bacteria can be modeled by a function,
N, which depends on the amount of time,
t (in seconds).
Before the medicine was introduced, there were 11,880 bacteria in the Petri dish.
Write a function that models the number of remaining bacteria
t seconds since the medicine was introduced.

N(t)=◻

After a special medicine is introduced into a petri dish containing a bacterial culture, the number of bacteria remaining in the dish decreases rapidly. $\newline$ The population loses $\frac{1}{4}$ of its size every $44$ seconds. The number of remaining bacteria can be modeled by a function, $N$ , which depends on the amount of time, $t$ (in seconds). $\newline$ Before the medicine was introduced, there were $11$ , $880$ bacteria in the Petri dish. $\newline$ Write a function that models the number of remaining bacteria $t$ seconds since the medicine was introduced. $\newline$ $N(t)=\square$

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Write exponential functions: word problems

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Q. After a special medicine is introduced into a petri dish containing a bacterial culture, the number of bacteria remaining in the dish decreases rapidly.

\newline

The population loses

\frac{1}{4}

of its size every

44

seconds. The number of remaining bacteria can be modeled by a function,

N

, which depends on the amount of time,

t

(in seconds).

\newline

Before the medicine was introduced, there were

11

,

880

bacteria in the Petri dish.

\newline

Write a function that models the number of remaining bacteria

t

seconds since the medicine was introduced.

\newline

N(t)=\square

Identify initial amount and rate: Identify the initial amount of bacteria and the rate of decrease. $\newline$ The initial amount of bacteria ( $a$ ) is given as $11,880$ . The bacteria population decreases by $\frac{1}{4}$ of its size every $44$ seconds, which means the remaining fraction of the population after each decrease is $\frac{3}{4}$ (since $1 - \frac{1}{4} = \frac{3}{4}$ ).
Determine decay factor: Determine the decay factor $b$ . $\newline$ The decay factor $b$ is the fraction of the population that remains after each time interval. In this case, every $44$ seconds, the population retains $\frac{3}{4}$ of its size. Therefore, $b = \frac{3}{4}$ .
Write exponential decay function: Write the exponential decay function. $\newline$ The general form of an exponential decay function is $N(t) = a(b)^t$ , where $N(t)$ is the number of bacteria at time $t$ , $a$ is the initial amount, $b$ is the decay factor, and $t$ is the time in the same units as the decay interval. However, since the decay happens every $44$ seconds, we need to adjust the exponent to reflect the number of $44$ -second intervals that have passed. This means we divide $t$ by $44$ to get the number of intervals.
Write final function: Write the final function. $\newline$ Using the values from the previous steps, the function that models the number of remaining bacteria $t$ seconds since the medicine was introduced is: $\newline$ $N(t) =$ $11$ , $880$ \times \left(\frac{ $3$ }{ $4$ }\right)^{\frac{t}{ $44$ }}

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