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)=

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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)=

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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 (1)/(2) kilogram of contaminants, and the purifier filters (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 (1)/(2) kilogram, and after each hour, (2)/(3) of the remaining contaminants are left. The function is C(t) = (initial amount) * (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) * (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) = (1/2) * (2/3)^t.

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