the half - life of a radioactive isotope is the time it takes for a quantity of the isotope to be reduced to…

the half - life of a radioactive isotope is the time it takes for a quantity of the isotope to be reduced to half its initial mass. starting with 180 grams of a radioactive isotope, how much will be left after 6 half - lives? use the calculator provided and round your answer to the nearest gram.

Answer

Explanation:

Step1: Identify decay formula

The formula for radioactive - decay is $A = A_0\times(\frac{1}{2})^n$, where $A_0$ is the initial amount, $n$ is the number of half - lives, and $A$ is the final amount.

Step2: Substitute values

Here, $A_0 = 180$ grams and $n = 6$. So, $A=180\times(\frac{1}{2})^6$.

Step3: Calculate $(\frac{1}{2})^6$

$(\frac{1}{2})^6=\frac{1}{2\times2\times2\times2\times2\times2}=\frac{1}{64}$.

Step4: Calculate final amount

$A = 180\times\frac{1}{64}=\frac{180}{64}=2.8125$ grams.

Step5: Round the answer

Rounding $2.8125$ to the nearest gram gives $3$ grams.