To solve for the amount of the sample of uranium-240 remaining after 9 hours and after 30 hours, we will use the given exponential decay function:
[tex]\[ A(t) = 1500 \left(\frac{1}{2}\right)^{\frac{t}{14}} \][/tex]
Let's break down the steps for each time interval:
### Step 1: Calculation for 9 hours
1. Identify the given values:
- Initial amount: [tex]\(1500\)[/tex] grams.
- Time [tex]\(t = 9\)[/tex] hours.
- Half-life of uranium-240: [tex]\(14\)[/tex] hours.
2. Substitute [tex]\(t = 9\)[/tex] into the decay function:
[tex]\[ A(9) = 1500 \left(\frac{1}{2}\right)^{\frac{9}{14}} \][/tex]
3. Calculate the exponent:
[tex]\[ \frac{9}{14} \approx 0.642857 \][/tex]
4. Evaluate the exponential term:
[tex]\[ \left(\frac{1}{2}\right)^{0.642857} \approx 0.64044 \][/tex]
5. Multiply by the initial amount:
[tex]\[ A(9) \approx 1500 \times 0.64044 = 960.665\][/tex]
6. Round to the nearest gram:
[tex]\[ \text{Amount after 9 hours} \approx 961 \text{ grams} \][/tex]
### Step 2: Calculation for 30 hours
1. Identify the given values:
- Initial amount: [tex]\(1500\)[/tex] grams.
- Time [tex]\(t = 30\)[/tex] hours.
- Half-life of uranium-240: [tex]\(14\)[/tex] hours.
2. Substitute [tex]\(t = 30\)[/tex] into the decay function:
[tex]\[ A(30) = 1500 \left(\frac{1}{2}\right)^{\frac{30}{14}} \][/tex]
3. Calculate the exponent:
[tex]\[ \frac{30}{14} \approx 2.142857 \][/tex]
4. Evaluate the exponential term:
[tex]\[ \left(\frac{1}{2}\right)^{2.142857} \approx 0.22643 \][/tex]
5. Multiply by the initial amount:
[tex]\[ A(30) \approx 1500 \times 0.22643 = 339.646\][/tex]
6. Round to the nearest gram:
[tex]\[ \text{Amount after 30 hours} \approx 340 \text{ grams} \][/tex]
### Final Answer
- The amount of uranium-240 remaining after 9 hours is approximately [tex]\(961\)[/tex] grams.
- The amount of uranium-240 remaining after 30 hours is approximately [tex]\(340\)[/tex] grams.
