Question

16.) the amount of miligrams of a certain medicine in a patients bloodstream ( t ) hours after administration is given by ( m(t) = 300e^{-2t} ).

a.) how many miligrams of the medicine were initially administered?

b.) how many miligrams will be in the patient’s bloodstream after 2 hours? (round to the nearest tenth.)

c.) find ( limlimits_{t \to infty} m(t) ). interpret the meaning of this limit.

Explanation:

Step1: Substitute $t=0$ for initial dose

$m(0) = 300e^{-2(0)} = 300e^0 = 300(1) = 300$

Step2: Substitute $t=2$ for 2-hour dose

$m(2) = 300e^{-2(2)} = 300e^{-4} \approx 300(0.0183) \approx 40.6$

Step3: Evaluate limit as $t\to\infty$

$\lim_{t \to \infty} 300e^{-2t} = 300\lim_{t \to \infty} \frac{1}{e^{2t}} = 300(0) = 0$
Interpret: As time approaches infinity, the medicine concentration in the bloodstream approaches 0, meaning the body fully eliminates the medicine over time.

Answer:

a.) 300 milligrams
b.) 40.6 milligrams
c.) $\lim_{t \to \infty} m(t) = 0$; Over time, the medicine is eliminated from the bloodstream.