$$ A(t)=2800\left(\frac{1}{2}\right)^{\frac{t}{14}} $$ Find the amount of the sample remaining after 11 hours and after 30 hours. Round your answers to the nearest gram as necessary. Amount after 11 hours: $$ \square $$ grams Amount after 30 hours: $$ \square $$ grams

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UpStudy AI · Accepted Answer

Given the decay formula

$$
A(t)=2800\left(\frac{1}{2}\right)^{\frac{t}{14}},
$$

we need to find the amount remaining after $$ t=11 $$ and $$ t=30 $$ hours.

### Amount after 11 hours

1. Substitute $$ t=11 $$:

$$
   A(11)=2800\left(\frac{1}{2}\right)^{\frac{11}{14}}.
   $$

2. Evaluate the exponent:

$$
   \frac{11}{14} \approx 0.7857.
   $$

3. Compute the expression:

$$
   \left(\frac{1}{2}\right)^{0.7857} = e^{0.7857 \ln\left(\frac{1}{2}\right)} = e^{-0.7857\ln(2)}.
   $$

Since $$\ln(2)\approx 0.6931$$:

$$
   -0.7857\ln(2) \approx -0.7857 \times 0.6931 \approx -0.5447,
   $$

so

$$
   e^{-0.5447}\approx 0.58045.
   $$

4. Multiply by 2800:

$$
   A(11)\approx 2800 \times 0.58045 \approx 1625.26.
   $$

5. Rounding to the nearest gram:

$$
   A(11)\approx 1625 \text{ grams}.
   $$

### Amount after 30 hours

1. Substitute $$ t=30 $$:

$$
   A(30)=2800\left(\frac{1}{2}\right)^{\frac{30}{14}}.
   $$

2. Simplify the exponent:

$$
   \frac{30}{14} = \frac{15}{7} \approx 2.1429.
   $$

3. Compute the expression:

$$
   \left(\frac{1}{2}\right)^{2.1429} = e^{2.1429 \ln\left(\frac{1}{2}\right)} = e^{-2.1429\ln(2)}.
   $$

With $$\ln(2)\approx 0.6931$$:

$$
   -2.1429\ln(2)\approx -2.1429\times 0.6931 \approx -1.4859,
   $$

so

$$
   e^{-1.4859}\approx 0.22644.
   $$

4. Multiply by 2800:

$$
   A(30)\approx 2800 \times 0.22644 \approx 633.99.
   $$

5. Rounding to the nearest gram:

$$
   A(30)\approx 634 \text{ grams}.
   $$

### Final Answers

Amount after 11 hours: $$1625$$ grams  
Amount after 30 hours: $$634$$ grams
After 11 hours, there are approximately 1625 grams remaining. After 30 hours, there are approximately 634 grams remaining.

Find the amount of the sample remaining after 11 hours and after 30 hours.
Round your answers to the nearest gram as necessary.
Amount after 11 hours: grams
Amount after 30 hours: grams

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Find the amount of the sample remaining after 11 hours and after 30 hours. Round your answers to the nearest gram as necessary. Amount after 11 hours: grams Amount after 30 hours: grams