Question

1. a doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour.

(a) to the nearest hour, what is the half - life of the drug?

Explanation:

Step1: Use the decay formula

The formula for exponential decay is \(A = A_0(1 - r)^t\), where \(A_0\) is the initial amount, \(r\) is the decay rate, and \(t\) is the time. For half - life, \(A=\frac{A_0}{2}\). Given \(r = 0.3\) (since it decays by 30% or \(0.3\) per hour). So, \(\frac{A_0}{2}=A_0(1 - 0.3)^t\).

Step2: Simplify the equation

Divide both sides of \(\frac{A_0}{2}=A_0(0.7)^t\) by \(A_0\) (since \(A_0
eq0\)), we get \(\frac{1}{2}=(0.7)^t\).

Step3: Take the natural logarithm of both sides

\(\ln(\frac{1}{2})=\ln(0.7^t)\). Using the property \(\ln(a^b)=b\ln(a)\), we have \(\ln(\frac{1}{2}) = t\ln(0.7)\).

Step4: Solve for \(t\)

\(t=\frac{\ln(\frac{1}{2})}{\ln(0.7)}\). We know that \(\ln(\frac{1}{2})=-\ln(2)\approx - 0.693\) and \(\ln(0.7)\approx-0.357\). Then \(t=\frac{- 0.693}{-0.357}\approx2\)

Answer:

The half - life of the drug is approximately \(2\) hours.