To determine how much of an element remains after it decays for a certain period, we use the concept of exponential decay. Here’s a step-by-step solution for finding the remaining mass of the element after 9 minutes, given a decay rate of [tex]\( 5.7 \% \)[/tex] per minute:
1. Initial Information:
- Initial mass = 310 grams
- Decay rate = [tex]\( 5.7 \% \)[/tex] per minute
- Time elapsed = 9 minutes
2. Decay Rate as a Decimal:
The decay rate of [tex]\( 5.7 \% \)[/tex] per minute can be converted to a decimal for use in calculations. Since [tex]\( 5.7\% = 0.057 \)[/tex] (as a decimal).
3. Exponential Decay Formula:
The general formula for exponential decay is:
[tex]\[
M(t) = M_0 \times (1 - r)^t
\][/tex]
where:
- [tex]\( M(t) \)[/tex] is the remaining mass after time [tex]\( t \)[/tex] minutes.
- [tex]\( M_0 \)[/tex] is the initial mass.
- [tex]\( r \)[/tex] is the decay rate per minute.
- [tex]\( t \)[/tex] is the time in minutes.
4. Substituting Values:
In this problem:
- [tex]\( M_0 = 310 \text{ grams} \)[/tex]
- [tex]\( r = 0.057 \)[/tex]
- [tex]\( t = 9 \text{ minutes} \)[/tex]
Thus, substituting these values into the formula:
[tex]\[
M(9) = 310 \times (1 - 0.057)^9
\][/tex]
5. Performing the Calculation:
First, calculate [tex]\( 1 - 0.057 \)[/tex]:
[tex]\[
1 - 0.057 = 0.943
\][/tex]
Then, raise this value to the power of 9:
[tex]\[
(0.943)^9
\][/tex]
Next, multiply the initial mass by this result:
[tex]\[
310 \times (0.943)^9
\][/tex]
6. Final Calculation:
After performing the above calculations, which involve applying the exponential decay function repeatedly over the 9-minute period, we get the remaining mass.
7. Conclusion:
The remaining mass of the element after 9 minutes is approximately:
[tex]\[
\boxed{182.8 \text{ grams}}
\][/tex]
This is rounded to the nearest 10th of a gram, as required by the problem statement.
