Answer:
The half-life of the substance is about 288 days.
Step-by-step explanation:
The exponential decay function:
[tex]\displaystyle A=A_0\left(\frac{1}{2}\right)^{t/P}[/tex]
Can determine the amount A of a radioactive substance present at time t. A₀ represents the initial amount and P is the half-life of the substance.
We are given that a substance loses 70% of its radioactivity in 500 days, and we want to determine the period of the half-life.
In other words, we want to determine P.
Since the substance has lost 70% of its radioactivity, it will have only 30% of its original amount. This occured in 500 days. Therefore, A = 0.3A₀ when t = 500 (days). Substitute:
[tex]\displaystyle 0.3A_0=A_0\left(\frac{1}{2}\right)^{500/P}[/tex]
Divide both sides by A₀:
[tex]\displaystyle 0.3=\left(\frac{1}{2}\right)^{500/P}[/tex]
We can take the natural log of both sides:
[tex]\displaystyle \ln(0.3)=\ln\left(\left(\frac{1}{2}\right)^{500/P}\right)[/tex]
Using logarithmic properties:
[tex]\displaystyle \ln(0.3)=\frac{500}{P}\left(\ln\left(\frac{1}{2}\right)\right)[/tex]
So:
[tex]\displaystyle \frac{500}{P}=\frac{\ln(0.3)}{\ln(0.5)}[/tex]
Take the reciprocal of both sides:
[tex]\displaystyle \frac{P}{500}=\displaystyle \frac{\ln(0.5)}{\ln(0.3)}[/tex]
Use a calculator:
[tex]\displaystyle P=\frac{500\ln(0.5)}{\ln(0.3)}\approx287.86[/tex]
The half-life of the substance is about 288 days.
