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At the moment a certain medicine is injected, its concentration in the bloodstream is 120 milligrams per liter. From that moment forward, the medicine's concentration drops by
30% each hour.
Write a function that gives the medicine's concentration in milligrams per liter,
C(t),t hours after the medicine was injected.

C(t)=◻++^(x)

At the moment a certain medicine is injected, its concentration in the bloodstream is $120$ milligrams per liter. From that moment forward, the medicine's concentration drops by $30 \%$ each hour. $\newline$ Write a function that gives the medicine's concentration in milligrams per liter, $C(t)$ , $t$ hours after the medicine was injected. $\newline$ $C(t)=\square$

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Interpret the exponential function word problem

Full solution

Q. At the moment a certain medicine is injected, its concentration in the bloodstream is

120

milligrams per liter. From that moment forward, the medicine's concentration drops by

30 \%

each hour.

\newline

Write a function that gives the medicine's concentration in milligrams per liter,

C(t)

,

t

hours after the medicine was injected.

\newline

C(t)=\square

Identify initial concentration and rate: Step $1$ : Identify the initial concentration and the rate of decrease. The initial concentration is given as $120$ milligrams per liter. The concentration decreases by $30\%$ each hour, which means that $70\%$ of the concentration remains each hour.
Convert percentage to decimal form: Step $2$ : Convert the percentage that remains each hour into decimal form to use as the base of the exponential function. Since $70\%$ remains, the decimal equivalent is $0.70$ .
Write exponential decay function: Step $3$ : Write the exponential decay function. The general form of an exponential decay function is $C(t) = C_0 \times b^t$ , where $C_0$ is the initial concentration and $b$ is the base representing the hourly decrease. In this case, $C_0 = 120$ and $b = 0.70$ .
Substitute values into function: Step $4$ : Substitute the values into the exponential decay function to get the specific function for this situation. $C(t) = 120 \times 0.70^t$ .
Verify function accuracy: Step $5$ : Verify that the function correctly represents the situation. At $t = 0$ (the moment the medicine is injected), $C(0) = 120 \times 0.70^0 = 120 \times 1 = 120$ milligrams per liter, which matches the initial condition.

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