Questions: A person is given an injection of 700 milligrams of penicillin at time t=0. Let f(t) be the amount (in milligrams) of penicillin present in the person's bloodstream t hours after the injection. Then, the amount of penicillin decays exponentially, and a typical formula is f(t)=700 e^(-0.8 t). Complete parts (a) through (c) below. (a) Give the differential equation satisfied by f(t). f'(t)=

A person is given an injection of 700 milligrams of penicillin at time t=0. Let f(t) be the amount (in milligrams) of penicillin present in the person's bloodstream t hours after the injection. Then, the amount of penicillin decays exponentially, and a typical formula is f(t)=700 e^(-0.8 t). Complete parts (a) through (c) below. (a) Give the differential equation satisfied by f(t). f'(t)=
Transcript text: A person is given an injection of 700 milligrams of penicillin at time $t=0$. Let $f(t)$ be the amount (in milligrams) of penicillin present in the person's bloodstream thours after the injection. Then, the amount of penicillin decays exponentially, and a typical formula is $\mathrm{f}(\mathrm{t})=700 e^{-0.8 t}$. Complete parts $(\mathrm{a})$ through ( c ) below. (a) Give the differential equation satisfied by $f(t)$. \[ f^{\prime}(t)=\square \]
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Solution

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Solution Steps

Step 1: Define the Function

The amount of penicillin present in the person's bloodstream at time \( t \) is given by the function: \[ f(t) = 700 e^{-0.8 t} \]

Step 2: Differentiate the Function

To find the rate of change of the amount of penicillin over time, we differentiate \( f(t) \) with respect to \( t \): \[ f'(t) = \frac{d}{dt}(700 e^{-0.8 t}) = -560 e^{-0.8 t} \]

Step 3: Write the Differential Equation

The differential equation that describes the rate of change of the amount of penicillin in the bloodstream is: \[ f'(t) = -560 e^{-0.8 t} \]

Final Answer

The differential equation satisfied by \( f(t) \) is: \[ \boxed{f'(t) = -560 e^{-0.8 t}} \]

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