(d) How much radium will be present in the sample in 100 years? 6. Radioactive Decay Carbon-14 is a radioactive element present in living organisms. After an organism dies, the amount $$ A $$ of carbon-14 present begins to decline at a rate proportional to the amount present at the time of death. The half-life of carbon-14 is 5730 years.

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To determine how much carbon-14 will be present in a sample after 100 years, we can use the formula for radioactive decay, which is given by:

$$
A(t) = A_0 e^{-kt}
$$

where:
- $$ A(t) $$ is the amount of substance remaining after time $$ t $$,
- $$ A_0 $$ is the initial amount of the substance,
- $$ k $$ is the decay constant,
- $$ t $$ is the time in years,
- $$ e $$ is the base of the natural logarithm.

### Step 1: Find the decay constant $$ k $$

The half-life of carbon-14 is given as 5730 years. The decay constant $$ k $$ can be calculated using the half-life formula:

$$
k = \frac{\ln(2)}{t_{1/2}}
$$

Substituting the half-life into the equation:

$$
k = \frac{\ln(2)}{5730}
$$

### Step 2: Calculate $$ k $$

Let's calculate $$ k $$ first.

$$
k = \frac{\ln(2)}{5730}
$$

### Step 3: Calculate the amount of carbon-14 after 100 years

Now, we can substitute $$ k $$ and $$ t = 100 $$ years into the decay formula.

$$
A(100) = A_0 e^{-k \cdot 100}
$$

### Step 4: Perform the calculations

I will calculate $$ k $$ and then use it to find $$ A(100) $$. For this calculation, we will assume an initial amount $$ A_0 = 1 $$ for simplicity, as the final result will be proportional to the initial amount.

Let's perform these calculations.
Simplify the expression by following steps:
- step0: Solution:
  $$e^{-k\times 100}$$
- step1: Reorder the terms:
  $$e^{-100k}$$
Calculate or simplify the expression $$ \ln(2)/5730 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\ln{\left(2\right)}}{5730}$$
Calculate or simplify the expression $$ e^(-0.000121*100) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$e^{-0.000121\times 100}$$
- step1: Multiply the numbers:
  $$e^{-0.0121}$$
- step2: Convert the expressions:
  $$e^{-\frac{121}{10000}}$$
- step3: Express with a positive exponent:
  $$\frac{1}{e^{\frac{121}{10000}}}$$
- step4: Simplify:
  $$\frac{1}{\sqrt[10000]{e^{121}}}$$
- step5: Multiply by the Conjugate:
  $$\frac{\sqrt[10000]{e^{9879}}}{\sqrt[10000]{e^{121}}\times \sqrt[10000]{e^{9879}}}$$
- step6: Multiply the numbers:
  $$\frac{\sqrt[10000]{e^{9879}}}{e}$$
### Step 1: Calculate the decay constant $$ k $$

We found that:

$$
k \approx 0.000121
$$

### Step 2: Calculate the amount of carbon-14 after 100 years

Now, substituting $$ k $$ into the decay formula:

$$
A(100) = A_0 e^{-0.000121 \cdot 100}
$$

This simplifies to:

$$
A(100) = A_0 e^{-0.0121}
$$

### Step 3: Calculate $$ A(100) $$

Now, we can calculate $$ e^{-0.0121} $$:

$$
A(100) \approx A_0 \cdot e^{-0.0121}
$$

Let's compute $$ e^{-0.0121} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$e^{-0.0121}$$
- step1: Convert the expressions:
  $$e^{-\frac{121}{10000}}$$
- step2: Express with a positive exponent:
  $$\frac{1}{e^{\frac{121}{10000}}}$$
- step3: Simplify:
  $$\frac{1}{\sqrt[10000]{e^{121}}}$$
- step4: Multiply by the Conjugate:
  $$\frac{\sqrt[10000]{e^{9879}}}{\sqrt[10000]{e^{121}}\times \sqrt[10000]{e^{9879}}}$$
- step5: Multiply the numbers:
  $$\frac{\sqrt[10000]{e^{9879}}}{e}$$
### Final Calculation

We found that:

$$
e^{-0.0121} \approx 0.987973
$$

### Conclusion

Thus, the amount of carbon-14 remaining after 100 years, assuming an initial amount $$ A_0 = 1 $$, is approximately:

$$
A(100) \approx 0.987973 \cdot A_0
$$

If you start with 1 unit of carbon-14, after 100 years, you will have approximately 0.987973 units of carbon-14 remaining.

If you have a specific initial amount $$ A_0 $$, you can multiply this result by that amount to find the exact quantity remaining.
After 100 years, approximately 0.988 units of carbon-14 remain in the sample.

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