Question
A 300 gram sample of thorium-234 has a half-life of 24 days. The mass of thorium in grams that remains after a certain time can be modeled using the function: m(t)=300(0.5)^t/24. What will the mass be after 31 days.
Answers
GPT-4o mini
To find the mass of thorium-234 after 31 days using the given function \( m(t) = 300(0.5)^{t/24} \), we will substitute \( t = 31 \) into the function:
\[
m(31) = 300(0.5)^{31/24}
\]
First, we need to calculate \( \frac{31}{24} \):
\[
\frac{31}{24} \approx 1.29167
\]
Now we calculate \( (0.5)^{1.29167} \):
\[
(0.5)^{1.29167} \approx 0.394
\]
Now, we can substitute this back into the mass function:
\[
m(31) = 300 \times 0.394
\]
Calculating the final mass:
\[
m(31) \approx 118.2 \text{ grams}
\]
Therefore, the mass of thorium-234 remaining after 31 days is approximately **118.2 grams**.
\[
m(31) = 300(0.5)^{31/24}
\]
First, we need to calculate \( \frac{31}{24} \):
\[
\frac{31}{24} \approx 1.29167
\]
Now we calculate \( (0.5)^{1.29167} \):
\[
(0.5)^{1.29167} \approx 0.394
\]
Now, we can substitute this back into the mass function:
\[
m(31) = 300 \times 0.394
\]
Calculating the final mass:
\[
m(31) \approx 118.2 \text{ grams}
\]
Therefore, the mass of thorium-234 remaining after 31 days is approximately **118.2 grams**.