The half-life of a certain element is 20 minutes. You weigh the element and record 80 grams. If you measure again 1.5 hours later, what has been the average amount of decay per minute?

A. 22.9
B. 0.85
C. 3.5



Answer :

To solve this problem, let's break it down into a step-by-step process:

1. Understand the given data:
- Initial weight of the element: [tex]\(80 \text{ grams}\)[/tex]
- Half-life of the element: [tex]\(20 \text{ minutes}\)[/tex]
- Time elapsed for the second measurement: [tex]\(1.5 \text{ hours}\)[/tex]

2. Convert the elapsed time to minutes:
[tex]\[ 1.5 \text{ hours} = 1.5 \times 60 \text{ minutes} = 90 \text{ minutes} \][/tex]

3. Determine how many half-lives have passed:
[tex]\[ \text{Number of half-lives passed} = \frac{\text{Elapsed time}}{\text{Half-life duration}} = \frac{90 \text{ minutes}}{20 \text{ minutes per half-life}} = 4.5 \text{ half-lives} \][/tex]

4. Calculate the remaining weight of the element using the half-life decay formula:
The remaining weight [tex]\(W\)[/tex] after [tex]\(n\)[/tex] half-lives is given by:
[tex]\[ W = W_0 \times \left( \frac{1}{2} \right)^n \][/tex]
Where:
- [tex]\(W_0 = 80 \text{ grams}\)[/tex] (initial weight)
- [tex]\(n = 4.5 \text{ half-lives}\)[/tex]

[tex]\[ W = 80 \text{ grams} \times \left( \frac{1}{2} \right)^{4.5} \approx 3.54 \text{ grams} \][/tex]

5. Calculate the total decay:
[tex]\[ \text{Total decay} = \text{Initial weight} - \text{Remaining weight} = 80 \text{ grams} - 3.54 \text{ grams} \approx 76.46 \text{ grams} \][/tex]

6. Calculate the average decay per minute:
[tex]\[ \text{Average decay per minute} = \frac{\text{Total decay}}{\text{Elapsed time in minutes}} = \frac{76.46 \text{ grams}}{90 \text{ minutes}} \approx 0.85 \text{ grams per minute} \][/tex]

Therefore, the average amount of decay per minute is:
[tex]\[ \boxed{0.85} \][/tex]