The half-life of polonium-218 is three minutes. How much of the isotope should you buy if you need a 0.10 g sample of this material, and you know the polonium-218 will take 30 minutes to reach you?

To determine how much polonium-218 you need to purchase to ensure you have a 0.10g sample after 30 minutes, follow these steps:

1. Understand the Concept of Half-life:
- The half-life of a substance is the time it takes for half of the material to decay. For polonium-218, this half-life is 3 minutes.

2. Determine the Number of Half-lives:
- Calculate how many half-lives will occur during the 30-minute journey.
- Since each half-life is 3 minutes, you divide the total travel time by the half-life:
[tex]\[
\text{Number of Half-lives} = \frac{\text{Travel Time}}{\text{Half-life}} = \frac{30}{3} = 10
\][/tex]

3. Calculate the Initial Amount Needed:
- To have a 0.10g sample after 30 minutes, calculate how much you need to start with.
- The formula to find the initial amount given the final amount and the number of half-lives is:
[tex]\[
\text{Initial Amount} = \text{Final Amount} \times (2^{\text{Number of Half-lives}})
\][/tex]
- Plug in the values:
[tex]\[
\text{Initial Amount} = 0.10 \, \text{g} \times (2^{10})
\][/tex]
- Calculate:
[tex]\[
2^{10} = 1024
\][/tex]
Therefore:
[tex]\[
\text{Initial Amount} = 0.10 \, \text{g} \times 1024 = 102.4 \, \text{g}
\][/tex]

So, you need to start with 102.4 grams of polonium-218 to ensure you have 0.10 grams after 30 minutes.