Question

Element X decays radioactively with a half-life of eight minutes. If there are 800 g of element X, how long, to the nearest 10th of a minute, would it take the element to decay 22g.

Answer 1

Answer: 41.5 min

Step-by-step explanation:

This problem can be solved with the Radioactive Half Life Formula:  

$A=A_{o}.2^{\frac{-t}{h}}$ (1)

Where:  

$A=22 g$ is the final amount of the radioactive element

$A_{o}=800 g$ is the initial amount of the radioactive element  

$t$ is the time elapsed  

$h=8 min$ is the half life of the radioactive element  

So, we need to substitute the given values and find $t$ from (1):  

$22 g=(800 g) 2^{\frac{-t}{8 min}}$ (2)  

$\frac{22 g}{800 g}=2^{\frac{-t}{8 min}}$ (3)  

$\frac{11}{400}=2^{\frac{-t}{8 min}}$ (4)  

Applying natural logarithm in both sides:  

$ln(\frac{11}{400})=ln(2^{\frac{-t}{8 min}})$ (5)  

$-3.593=-\frac{t}{8 min}ln(2)$ (6)  

Clearing $t$:

$t=41.46 min \approx 41.5 min$ This is the time elapsed