To find the amount of the radioactive substance uranium-240 remaining after a certain number of hours, we use the given exponential decay formula:
[tex]\[
A(t) = 3900 \left( \frac{1}{2} \right)^{\frac{t}{14}}
\][/tex]
We need to calculate the amount remaining after two different time periods: 11 hours and 60 hours.
### Step-by-Step Solution
1. Amount remaining after 11 hours:
First, we substitute [tex]\( t = 11 \)[/tex] into the formula:
[tex]\[
A(11) = 3900 \left( \frac{1}{2} \right)^{\frac{11}{14}}
\][/tex]
Evaluate the exponent:
[tex]\[
\frac{11}{14} \approx 0.7857
\][/tex]
Now, calculate the base raised to this power:
[tex]\[
\left( \frac{1}{2} \right)^{\frac{11}{14}} \approx 0.5808
\][/tex]
Finally, multiply this by the initial amount (3900 grams):
[tex]\[
A(11) = 3900 \times 0.5808 \approx 2262
\][/tex]
So, the amount of the sample remaining after 11 hours is approximately 2262 grams.
2. Amount remaining after 60 hours:
Now, substitute [tex]\( t = 60 \)[/tex] into the formula:
[tex]\[
A(60) = 3900 \left( \frac{1}{2} \right)^{\frac{60}{14}}
\][/tex]
Evaluate the exponent:
[tex]\[
\frac{60}{14} \approx 4.2857
\][/tex]
Next, calculate the base raised to this power:
[tex]\[
\left( \frac{1}{2} \right)^{4.2857} \approx 0.0513
\][/tex]
Multiply this by the initial amount (3900 grams):
[tex]\[
A(60) = 3900 \times 0.0513 \approx 200
\][/tex]
So, the amount of the sample remaining after 60 hours is approximately 200 grams.
### Answers
- Amount after 11 hours: [tex]\( 2262 \)[/tex] grams
- Amount after 60 hours: [tex]\( 200 \)[/tex] grams
These rounded values provide the solution to the given problem regarding the remaining amount of uranium-240.
