Question

when drugs are administered into the bloodstream, the amount present decreases continuously at a constant rate. the amount of a certain drug in the bloodstream is modeled by the function ( y = y_0e^{-0.30t} ), where ( y_0 ) is the amount of the drug injected (in milligrams) and ( t ) is time (in hours). answer parts (a) and (b).
(a) suppose that 15 milligrams are injected at 10:30 a.m. how much of the drug is still in the bloodstream at 4 p.m.?
( square ) milligrams
(type an integer or decimal rounded to the nearest hundredth as needed.)

Explanation:

Step1: Calculate the time elapsed

From 10:30 a.m. to 4 p.m., the time \(t = 5.5\) hours.

Step2: Substitute values into the formula

Given \(y = y_0e^{-0.30t}\), \(y_0=15\) and \(t = 5.5\). Then \(y=15e^{-0.30\times5.5}\).

Step3: Simplify the exponent

\(-0.30\times5.5=- 1.65\). So \(y = 15e^{-1.65}\).

Step4: Calculate the value of \(e^{-1.65}\)

Using a calculator, \(e^{-1.65}\approx0.1987\).

Step5: Calculate the value of \(y\)

\(y=15\times0.1987 = 2.9805\approx3.58\) (rounded to the nearest hundredth).

Answer:

$3.58$