Sure, let's go through the problem step-by-step.
We start with an initial mass of 780 grams. The decay rate per minute is 16.3%, which means that each minute, 16.3% of the mass is lost. This can also be understood as retaining 100% - 16.3% = 83.7% of the mass each minute.
Since we are dealing with exponential decay, the mass remaining after a certain number of minutes can be calculated using the formula for exponential decay:
[tex]\[ m(t) = m_0 \cdot (1 - \text{decay\_rate})^t \][/tex]
Where:
- [tex]\( m(t) \)[/tex] is the mass remaining after [tex]\( t \)[/tex] minutes,
- [tex]\( m_0 \)[/tex] is the initial mass,
- decay\_rate is the rate of decay per minute in decimal form,
- [tex]\( t \)[/tex] is the number of minutes.
### Step-by-Step Calculation:
1. Initial Mass ([tex]\( m_0 \)[/tex]):
- [tex]\( m_0 = 780 \)[/tex] grams.
2. Decay Rate (as a decimal):
- Decay rate per minute is 16.3%, which is [tex]\( 0.163 \)[/tex] in decimal form.
3. Number of Minutes ([tex]\( t \)[/tex]):
- [tex]\( t = 16 \)[/tex] minutes.
4. Formula Substitution:
[tex]\[
m(16) = 780 \cdot (1 - 0.163)^{16}
\][/tex]
[tex]\[
m(16) = 780 \cdot (0.837)^{16}
\][/tex]
5. Calculate the Remaining Mass:
[tex]\[
(0.837)^{16} \approx 0.058
\][/tex]
[tex]\[
m(16) = 780 \cdot 0.058 \approx 45.26 \text{ grams}
\][/tex]
6. Round to the Nearest 10th:
- Rounding 45.26 grams to the nearest 10th gives us approximately 45.3 grams.
### Conclusion:
After 16 minutes, the remaining mass of the element, rounded to the nearest 10th of a gram, is approximately 45.3 grams.
