The half-life of a radioactive kind of tin is 10 days. If you start with 66,048 grams of it, how much will be left after 50 days?

Steps to Solve

1. Calculate the number of half-lives To find how many half-lives have passed in 50 days, divide the total time (50 days) by the length of one half-life (10 days).

$$ \text{Number of half-lives} = \frac{50 \text{ days}}{10 \text{ days}} = 5 $$

2. Apply the half-life formula The remaining amount of a substance after a certain number of half-lives can be calculated using the formula:

$$ \text{Remaining amount} = \text{Initial amount} \left( \frac{1}{2} \right)^{n} $$

where ( n ) is the number of half-lives. Here, ( n = 5 ) and the initial amount is 66,048 grams.

3. Calculate the remaining amount Substituting the values into the formula:

$$ \text{Remaining amount} = 66,048 \left( \frac{1}{2} \right)^{5} $$

4. Simplify the remaining amount calculation Calculate ( \left( \frac{1}{2} \right)^{5} ):

$$ \left( \frac{1}{2} \right)^{5} = \frac{1}{32} $$

Then plug this back into the equation:

$$ \text{Remaining amount} = 66,048 \times \frac{1}{32} $$

5. Final calculation Now perform the multiplication:

$$ \text{Remaining amount} = 66,048 \div 32 = 2,064 $$