Discover a wealth of knowledge and get your questions answered at IDNLearn.com. Join our Q&A platform to access reliable and detailed answers from experts in various fields.
Sagot :
To find the mass of a radioactive substance decaying at an exponential rate of 2% per day, we need to use the 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 mass at time [tex]\( t \)[/tex],
- [tex]\( M_0 \)[/tex] is the initial mass,
- [tex]\( r \)[/tex] is the decay rate,
- [tex]\( t \)[/tex] is the time in days.
Given the problem:
- The initial mass [tex]\( M_0 \)[/tex] is 80 grams,
- The decay rate [tex]\( r \)[/tex] is 2%, which can be written as 0.02,
- The time [tex]\( t \)[/tex] is 5 days.
Let's plug these values into the formula:
1. Identify the initial mass [tex]\( M_0 \)[/tex]:
[tex]\[ M_0 = 80 \text{ grams} \][/tex]
2. Identify the decay rate [tex]\( r \)[/tex]:
[tex]\[ r = 0.02 \][/tex]
3. Identify the time in days [tex]\( t \)[/tex]:
[tex]\[ t = 5 \text{ days} \][/tex]
Now, substitute these values into the exponential decay formula:
[tex]\[ M(5) = 80 \times (1 - 0.02)^5 \][/tex]
We need to calculate [tex]\( (1 - 0.02)^5 \)[/tex]:
[tex]\[ (1 - 0.02) = 0.98 \][/tex]
Then raise 0.98 to the power of 5:
[tex]\[ 0.98^5 \approx 0.9043829759 \][/tex]
Finally, multiply the initial mass by this value:
[tex]\[ M(5) = 80 \times 0.9043829759 \][/tex]
[tex]\[ M(5) \approx 72.313663744 \][/tex]
So, the mass of the radioactive substance at the end of 5 days is approximately 72.31 grams (rounded to two decimal places).
[tex]\[ M(t) = M_0 \times (1 - r)^t \][/tex]
where:
- [tex]\( M(t) \)[/tex] is the mass at time [tex]\( t \)[/tex],
- [tex]\( M_0 \)[/tex] is the initial mass,
- [tex]\( r \)[/tex] is the decay rate,
- [tex]\( t \)[/tex] is the time in days.
Given the problem:
- The initial mass [tex]\( M_0 \)[/tex] is 80 grams,
- The decay rate [tex]\( r \)[/tex] is 2%, which can be written as 0.02,
- The time [tex]\( t \)[/tex] is 5 days.
Let's plug these values into the formula:
1. Identify the initial mass [tex]\( M_0 \)[/tex]:
[tex]\[ M_0 = 80 \text{ grams} \][/tex]
2. Identify the decay rate [tex]\( r \)[/tex]:
[tex]\[ r = 0.02 \][/tex]
3. Identify the time in days [tex]\( t \)[/tex]:
[tex]\[ t = 5 \text{ days} \][/tex]
Now, substitute these values into the exponential decay formula:
[tex]\[ M(5) = 80 \times (1 - 0.02)^5 \][/tex]
We need to calculate [tex]\( (1 - 0.02)^5 \)[/tex]:
[tex]\[ (1 - 0.02) = 0.98 \][/tex]
Then raise 0.98 to the power of 5:
[tex]\[ 0.98^5 \approx 0.9043829759 \][/tex]
Finally, multiply the initial mass by this value:
[tex]\[ M(5) = 80 \times 0.9043829759 \][/tex]
[tex]\[ M(5) \approx 72.313663744 \][/tex]
So, the mass of the radioactive substance at the end of 5 days is approximately 72.31 grams (rounded to two decimal places).
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. Your questions deserve accurate answers. Thank you for visiting IDNLearn.com, and see you again for more solutions.