Answer :
To solve this problem, we need to use the concept of half-life, which is the time required for a quantity to reduce to half its initial value.
### Steps to solve the problem:
1. Understand the half-life formula:
The formula that relates the initial amount, final amount, half-life, and time elapsed is:
[tex]\[
\text{Final amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^{\frac{\text{time elapsed}}{\text{half-life}}}
\][/tex]
2. Identify the given values:
- Initial amount of palladium-103 ([tex]\(A_0\)[/tex]) = 16 grams
- Final amount of palladium-103 ([tex]\(A_t\)[/tex]) = 1.0 gram
- Half-life of palladium-103 ([tex]\(t_{1/2}\)[/tex]) = 17 days
3. Set up the formula:
Substitute the given values into the half-life decay formula:
[tex]\[
1 = 16 \times \left(\frac{1}{2}\right)^{\frac{t}{17}}
\][/tex]
4. Solve for [tex]\(t\)[/tex]:
To isolate [tex]\(t\)[/tex], first divide both sides of the equation by 16:
[tex]\[
\frac{1}{16} = \left(\frac{1}{2}\right)^{\frac{t}{17}}
\][/tex]
5. Convert to logarithmic form:
Take the logarithm base 2 of both sides to simplify the exponent:
[tex]\[
\log_2\left(\frac{1}{16}\right) = \frac{t}{17}
\log_2\left(\frac{1}{2}\right)
\][/tex]
Recall that [tex]\(\log_2\left(\frac{1}{2}\right) = -1\)[/tex], so:
[tex]\[
\log_2\left(\frac{1}{16}\right) = -\frac{t}{17}
\][/tex]
6. Calculate [tex]\(\log_2\left(\frac{1}{16}\right)\)[/tex]:
[tex]\(\frac{1}{16}\)[/tex] is the same as [tex]\(2^{-4}\)[/tex], so:
[tex]\[
\log_2\left(2^{-4}\right) = -4
\][/tex]
7. Solve for [tex]\(t\)[/tex]:
[tex]\[
-4 = -\frac{t}{17}
\][/tex]
Multiply both sides by -17 to solve for [tex]\(t\)[/tex]:
[tex]\[
t = 68 \text{ days}
\][/tex]
### Conclusion:
It takes 68 days for 16 grams of palladium-103 to decay to 1.0 gram.
### Steps to solve the problem:
1. Understand the half-life formula:
The formula that relates the initial amount, final amount, half-life, and time elapsed is:
[tex]\[
\text{Final amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^{\frac{\text{time elapsed}}{\text{half-life}}}
\][/tex]
2. Identify the given values:
- Initial amount of palladium-103 ([tex]\(A_0\)[/tex]) = 16 grams
- Final amount of palladium-103 ([tex]\(A_t\)[/tex]) = 1.0 gram
- Half-life of palladium-103 ([tex]\(t_{1/2}\)[/tex]) = 17 days
3. Set up the formula:
Substitute the given values into the half-life decay formula:
[tex]\[
1 = 16 \times \left(\frac{1}{2}\right)^{\frac{t}{17}}
\][/tex]
4. Solve for [tex]\(t\)[/tex]:
To isolate [tex]\(t\)[/tex], first divide both sides of the equation by 16:
[tex]\[
\frac{1}{16} = \left(\frac{1}{2}\right)^{\frac{t}{17}}
\][/tex]
5. Convert to logarithmic form:
Take the logarithm base 2 of both sides to simplify the exponent:
[tex]\[
\log_2\left(\frac{1}{16}\right) = \frac{t}{17}
\log_2\left(\frac{1}{2}\right)
\][/tex]
Recall that [tex]\(\log_2\left(\frac{1}{2}\right) = -1\)[/tex], so:
[tex]\[
\log_2\left(\frac{1}{16}\right) = -\frac{t}{17}
\][/tex]
6. Calculate [tex]\(\log_2\left(\frac{1}{16}\right)\)[/tex]:
[tex]\(\frac{1}{16}\)[/tex] is the same as [tex]\(2^{-4}\)[/tex], so:
[tex]\[
\log_2\left(2^{-4}\right) = -4
\][/tex]
7. Solve for [tex]\(t\)[/tex]:
[tex]\[
-4 = -\frac{t}{17}
\][/tex]
Multiply both sides by -17 to solve for [tex]\(t\)[/tex]:
[tex]\[
t = 68 \text{ days}
\][/tex]
### Conclusion:
It takes 68 days for 16 grams of palladium-103 to decay to 1.0 gram.
