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Raymond works for a pharmaceutical company that is testing the effectiveness of a new medication. He gave one of the study participants $400$ milligrams of the medication, and he knows the amount remaining in the participant's body will decrease by a factor of $1/3$ each hour. $\newline$ Write an exponential equation in the form $y = a(b)^x$ that can model the amount of medication, $y$ , remaining in the participant's body after $x$ hours. $\newline$ Use whole numbers, decimals, or simplified fractions for the values of $a$ and $b$ . $\newline$ $y =$ ______ $\newline$

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Write exponential functions: word problems

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Q. Raymond works for a pharmaceutical company that is testing the effectiveness of a new medication. He gave one of the study participants

400

milligrams of the medication, and he knows the amount remaining in the participant's body will decrease by a factor of

1/3

each hour.

\newline

Write an exponential equation in the form

y = a(b)^x

that can model the amount of medication,

y

, remaining in the participant's body after

x

hours.

\newline

Use whole numbers, decimals, or simplified fractions for the values of

a

and

b

.

\newline

y =

______

\newline

Identify Initial Amount: Identify the initial amount of medication given to the participant. The initial amount of medication $a$ is the starting quantity before any decrease occurs. $a = 400$ milligrams
Determine Decay Factor: Determine the decay factor $b$ . Since the medication decreases by a factor of $\frac{1}{3}$ each hour, the remaining amount is $\frac{2}{3}$ of the previous amount each hour (because $1 - \frac{1}{3} = \frac{2}{3}$ ). Therefore, the decay factor $b$ is $\frac{2}{3}$ .
Write Decay Equation: Write the exponential decay equation. $\newline$ The general form of an exponential decay equation is $y = a(b)^x$ . $\newline$ Substitute the values of $a$ and $b$ into the equation. $\newline$ $a = 400$ $\newline$ $b = \frac{2}{3}$ $\newline$ So, the equation is $y = 400\left(\frac{2}{3}\right)^x$ .

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