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]
