Let's go through the process of finding the half-life of a radioactive substance, given a decay rate of [tex]\(2.8\%\)[/tex] per year.
1. Understand the Decay Rate:
The given decay rate is 2.8% per year. This can be expressed as a decimal:
[tex]\[
k = -0.028
\][/tex]
The negative sign indicates that the substance is decaying over time.
2. Half-Life Formula:
The half-life ([tex]\(T\)[/tex]) of a radioactive substance is the time it takes for half of the substance to decay. It can be calculated using the formula:
[tex]\[
T = \frac{\ln(2)}{k}
\][/tex]
where [tex]\( \ln(2) \)[/tex] is the natural logarithm of 2, approximately equal to 0.693.
3. Calculate the Half-Life:
Plug in the values:
[tex]\[
T = \frac{0.693}{-0.028}
\][/tex]
Performing the division gives us the exact half-life value.
4. Exact Half-Life:
The exact value, when calculated, is:
[tex]\[
T \approx -24.755256448569472 \text{ years}
\][/tex]
Note that the negative sign aligns with the negative decay rate, but for the purpose of interpreting the half-life as a duration, we discard the negative sign:
[tex]\[
T \approx 24.755256448569472 \text{ years}
\][/tex]
5. Round to One Decimal Place:
We are asked to round the half-life to one decimal place:
[tex]\[
T \approx 24.8 \text{ years}
\][/tex]
Thus, the half-life of the radioactive substance, when the decay rate is 2.8% per year, is approximately 24.8 years.
