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Nana has a water purifier that filters
(1)/(3) of the contaminants each hour. She used it to purify water that had
(1)/(2) kilogram of contaminants.
Write a function that gives the remaining amount of contaminants in kilograms,
C(t),t hours after Nana started purifying the water.

C(t)=

Nana has a water purifier that filters $\frac{1}{3}$ of the contaminants each hour. She used it to purify water that had $\frac{1}{2}$ kilogram of contaminants. $\newline$ Write a function that gives the remaining amount of contaminants in kilograms, $C(t), t$ hours after Nana started purifying the water. $\newline$ $C(t)= \square$

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Interpret the exponential function word problem

Full solution

Q. Nana has a water purifier that filters

\frac{1}{3}

of the contaminants each hour. She used it to purify water that had

\frac{1}{2}

kilogram of contaminants.

\newline

Write a function that gives the remaining amount of contaminants in kilograms,

C(t), t

hours after Nana started purifying the water.

\newline

C(t)= \square

Identify Contaminants and Rate: Step $1$ : Identify the initial amount of contaminants and the rate at which the purifier filters the contaminants. Nana starts with $\frac{1}{2}$ kilogram of contaminants, and the purifier filters $\frac{1}{3}$ of the contaminants each hour.
Calculate Remaining Contaminants: Step $2$ : Since the purifier filters $(1)/(3)$ of the contaminants each hour, this means that each hour, $(2)/(3)$ of the contaminants remain. The function for the remaining contaminants will be a geometric decay function.
Write Remaining Contaminants Function: Step $3$ : Write the function for the remaining amount of contaminants. The initial amount is $(\frac{1}{2})$ kilogram, and after each hour, $(\frac{2}{3})$ of the remaining contaminants are left. The function is $C(t) = (\text{initial amount}) \times (\text{remaining fraction})^t$ .
Substitute Values into Function: Step $4$ : Substitute the known values into the function. The initial amount is $(1)/(2)$ kilogram, and the remaining fraction each hour is $(2)/(3)$ . Therefore, the function is $C(t) = (1/2) \times (2/3)^t$ .
Simplify Final Function: Step $5$ : Simplify the function if necessary. In this case, the function is already in its simplest form. So, the final function is $C(t) = (\frac{1}{2}) \cdot (\frac{2}{3})^t$ .

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