Question

a doctor prescribes 325 milligrams of a therapeutic drug that decays by about 15% each hour. to the nearest hour, what is the half - life of the drug? question help: video

Explanation:

Step1: Set up the decay formula

The general formula for exponential decay is $A = A_0(1 - r)^t$, where $A_0$ is the initial amount, $r$ is the rate of decay, and $t$ is the time. Here, $A_0=325$, $r = 0.15$, and we want to find $t$ when $A=\frac{A_0}{2}$. So the equation becomes $\frac{A_0}{2}=A_0(1 - 0.15)^t$.

Step2: Simplify the equation

Divide both sides of the equation $\frac{A_0}{2}=A_0(0.85)^t$ by $A_0$ (since $A_0
eq0$), we get $\frac{1}{2}=(0.85)^t$.

Step3: Take the natural - logarithm of both sides

$\ln(\frac{1}{2})=\ln(0.85^t)$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we have $\ln(\frac{1}{2}) = t\ln(0.85)$.

Step4: Solve for $t$

We know that $\ln(\frac{1}{2})=-\ln(2)\approx - 0.693$ and $\ln(0.85)\approx- 0.1625$. Then $t=\frac{\ln(\frac{1}{2})}{\ln(0.85)}=\frac{-0.693}{-0.1625}\approx4.26$.

Answer:

4