The half-life of an isotope is 1000 years. A $50$-gram sample of the isotope is sealed in a box.

1. How much is left after 10,000 years?

   Hint (a):

   Half-life means time in which half of the original $50$ grams will remain.

   More Help (a):

   $y = ab^{n}$a is the initial value,
   b is the multiplier ($0.5$ since we are working with a half-life),
   n is the number of thousands of years, and
   y is the amount of the isotope that remains (in grams).

   The equation is $y = 50\left(0.5\right)^{x}$. Substitute $10$ for $x$.

   Alternate Method (a):

   Use the equation below, where x is the number of years.

   $y = 50(0.5)^{\frac{x}{1000}}$

2. How long will it take to reduce to $1\%$ of the original amount?

   Hint (b):

   Use the equation from the help for part (a).
   In this case, $y = 1\%$ of $50$ and you are solving for x.

   Answer (b):

   $≈ 6600 \text{ years}$

3. How long will it take until all of the original sample of the isotope is gone? Support your answer.

   Hint (c):

   $0=50\left( \frac{1}{2}\right) ^x$