Question

a tumor is injected with 2.5 grams of iodine, which has a decay rate of 1.15% per day. a. to the nearest day, how long will it take for half of the iodine to decay? days b. if we start with 123,456,789 grams of iodine, to the nearest day, how long will it take for half of the iodine to decay? days c. if we start with 0.00000000123456789 grams of iodine, to the nearest day, how long will it take for half of the iodine to decay? days. question help: video

Explanation:

Step1: Identify the decay formula

The formula for exponential decay is $A = A_0(1 - r)^t$, where $A$ is the final amount, $A_0$ is the initial amount, $r$ is the decay rate, and $t$ is the time. We want to find the time $t$ when $A=\frac{A_0}{2}$, and $r = 0.0115$. So, $\frac{A_0}{2}=A_0(1 - 0.0115)^t$.

Step2: Simplify the equation

Divide both sides of the equation $\frac{A_0}{2}=A_0(0.9885)^t$ by $A_0$ (since $A_0
eq0$), we get $\frac{1}{2}=(0.9885)^t$.

Step3: Take the natural - logarithm of both sides

$\ln(\frac{1}{2})=\ln(0.9885^t)$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we have $\ln(\frac{1}{2}) = t\ln(0.9885)$.

Step4: Solve for $t$

$t=\frac{\ln(\frac{1}{2})}{\ln(0.9885)}=\frac{-\ln(2)}{\ln(0.9885)}$.

Step5: Calculate the value of $t$

We know that $\ln(2)\approx0.6931$ and $\ln(0.9885)\approx - 0.0116$. Then $t=\frac{0.6931}{0.0116}\approx60$. Since the time $t$ to reach the half - life does not depend on the initial amount of iodine, the answer for all parts (a), (b), and (c) is the same.

Answer:

60
60
60