To solve this problem, we will use the given exponential decay function for uranium-240:
[tex]\[ A(t) = 1500 \left(\frac{1}{2}\right)^\frac{t}{14} \][/tex]
### Step 1: Determine the Initial Amount
The initial amount is the amount of the substance at time [tex]\( t = 0 \)[/tex] hours. By substituting [tex]\( t = 0 \)[/tex] into the function, we get:
[tex]\[ A(0) = 1500 \left(\frac{1}{2}\right)^\frac{0}{14} \][/tex]
Since any number raised to the power of 0 is 1:
[tex]\[ A(0) = 1500 \times 1 = 1500 \][/tex]
Therefore, the initial amount is 1500 grams.
### Step 2: Determine the Amount Remaining After 50 Hours
To find the amount remaining after 50 hours, we substitute [tex]\( t = 50 \)[/tex] into the function:
[tex]\[ A(50) = 1500 \left(\frac{1}{2}\right)^\frac{50}{14} \][/tex]
We simplify the exponent:
[tex]\[ \frac{50}{14} \approx 3.571 \][/tex]
So the function becomes:
[tex]\[ A(50) = 1500 \left(\frac{1}{2}\right)^{3.571} \][/tex]
Now, we calculate [tex]\( \left(\frac{1}{2}\right)^{3.571} \)[/tex]:
[tex]\[ \left(\frac{1}{2}\right)^{3.571} \approx 0.084 \][/tex]
Finally, multiplying this by the initial amount:
[tex]\[ A(50) = 1500 \times 0.084 \approx 126 \][/tex]
Therefore, the amount remaining after 50 hours is 126 grams.
### Conclusion
- Initial amount: 1500 grams
- Amount after 50 hours: 126 grams
