The radioactive substance uranium-240 has a half-life of 14 hours. The amount [tex]\(A(t)\)[/tex] of a sample of uranium-240 remaining (in grams) after [tex]\(t\)[/tex] hours is given by the following exponential function:
[tex]\[ A(t)=1500\left(\frac{1}{2}\right)^{\frac{t}{14}} \][/tex]

Find the initial amount in the sample and the amount remaining after 50 hours. Round your answers to the nearest gram as necessary.

Initial amount: [tex]\(\_\_\_\_\)[/tex] grams

Amount after 50 hours: [tex]\(\_\_\_\_\)[/tex] grams



Answer :

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