Question

a doctor prescribes 325 milligrams of a therapeutic drug that decays by about 25% each hour. write an exponential model representing the amount of the remaining in the patients system after t hours. find the amount of the drug that would remain in the patients system after 4 hours. round to the nearest milligram. model: a(t)= remining after 4 hours: milligrams question help: video written example

Explanation:

Step1: Identify the initial amount and decay factor

The initial amount of the drug $a = 325$ mg. The decay rate is $25\%=0.25$, so the decay - factor $b=1 - 0.25 = 0.75$. The general form of an exponential decay model is $A(t)=a\cdot b^{t}$.

Step2: Write the exponential model

Substitute $a = 325$ and $b = 0.75$ into the formula, we get $A(t)=325\cdot(0.75)^{t}$.

Step3: Calculate the amount after 4 hours

Substitute $t = 4$ into the model $A(t)=325\cdot(0.75)^{t}$. So $A(4)=325\cdot(0.75)^{4}$. First, calculate $(0.75)^{4}=0.75\times0.75\times0.75\times0.75 = 0.31640625$. Then $A(4)=325\times0.31640625\approx103$ mg.

Answer:

Model: $A(t)=325\cdot(0.75)^{t}$
Remaining after 4 hours: 103