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).
