Sure! Let's work through this problem step-by-step.
### Step 1: Understand the Problem
You are given:
- The half-life of uranium-238, which is [tex]\(4.46 \times 10^9\)[/tex] years.
- The initial mass of uranium-238 in the sample, which is 4.0 milligrams.
- The time that has passed, which is also [tex]\(4.46 \times 10^9\)[/tex] years.
### Step 2: Determine the Number of Half-Lives
The half-life is the amount of time it takes for half of a radioactive substance to decay. To find out how many half-lives have passed, you divide the total time passed by the half-life.
[tex]\[
\text{Number of Half-Lives} = \frac{\text{Time Passed}}{\text{Half-Life}} = \frac{4.46 \times 10^9 \text{ years}}{4.46 \times 10^9 \text{ years}} = 1
\][/tex]
### Step 3: Calculate the Remaining Mass
To find the remaining mass of uranium-238 after a certain number of half-lives, you use the formula:
[tex]\[
\text{Remaining Mass} = \text{Initial Mass} \times \left(\frac{1}{2}\right)^{\text{Number of Half-Lives}}
\][/tex]
Since 1 half-life has passed:
[tex]\[
\text{Remaining Mass} = 4.0 \text{ mg} \times \left(\frac{1}{2}\right)^1 = 4.0 \text{ mg} \times 0.5 = 2.0 \text{ mg}
\][/tex]
Therefore, after [tex]\(4.46 \times 10^9\)[/tex] years, the sample would contain 2.0 milligrams of uranium-238.
