To solve the problem of finding the initial amount and the remaining amount of uranium-240 after 60 hours, let's break it down step-by-step using the given exponential function.
The function is:
[tex]\[ A(t) = 4700 \left(\frac{1}{2}\right)^{\frac{t}{14}} \][/tex]
### Step 1: Finding the Initial Amount
The initial amount is the amount when time [tex]\( t = 0 \)[/tex].
Substitute [tex]\( t = 0 \)[/tex] into the function:
[tex]\[ A(0) = 4700 \left(\frac{1}{2}\right)^{\frac{0}{14}} \][/tex]
Simplify the exponent:
[tex]\[ A(0) = 4700 \left(\frac{1}{2}\right)^0 \][/tex]
Any number raised to the power of 0 is 1:
[tex]\[ A(0) = 4700 \cdot 1 \][/tex]
So, the initial amount is:
[tex]\[ \boxed{4700} \, \text{grams} \][/tex]
### Step 2: Finding the Amount Remaining After 60 Hours
Now, we need to find the amount remaining after [tex]\( t = 60 \)[/tex] hours.
Substitute [tex]\( t = 60 \)[/tex] into the function:
[tex]\[ A(60) = 4700 \left(\frac{1}{2}\right)^{\frac{60}{14}} \][/tex]
First, calculate the exponent:
[tex]\[ \frac{60}{14} \approx 4.2857 \][/tex]
Then, evaluate the exponential expression [tex]\(\left(\frac{1}{2}\right)^{4.2857}\)[/tex]:
[tex]\[ \left(\frac{1}{2}\right)^{4.2857} \approx 0.0513 \][/tex]
Now multiply this value by the initial amount:
[tex]\[ A(60) = 4700 \cdot 0.0513 \approx 240.97 \][/tex]
When rounded to the nearest gram, we get:
[tex]\[ A(60) \approx 241 \, \text{grams} \][/tex]
So, the amount remaining after 60 hours is:
[tex]\[ \boxed{241} \, \text{grams} \][/tex]
### Final Answers
- Initial amount: [tex]\(\boxed{4700} \, \text{grams}\)[/tex]
- Amount after 60 hours: [tex]\(\boxed{241} \, \text{grams}\)[/tex]
This detailed step-by-step explanation ensures proper understanding of how the initial and remaining amounts are computed using the given exponential decay function.
