When a certain radioactive elements decays, the amount at any time t can be calculated using the function E(t)=`ae^((-t)/(500))` , where a is the original amound an t is the elapsed time in year.How many years would it take for an initial amount of 250 miligrams of this element to decay to 100 miligrams?

Options

A. 125 years
B. 200 years
C. 458 years
D. 496 years

Explanation

To solve the problem, we need to determine how many years it would take for an initial amount of 250 milligrams of a radioactive element to decay to 100 milligrams using the decay function provided.

### Step-by-Step Solution:

1. **Identify the decay function**:
The amount of the radioactive element at time \( t \) is given by the function:
\[ E(t) = ae^{-\frac{t}{500}} \]
where \( a \) is the initial amount and \( t \) is the time in years.

2. **Substitute the known values**:
We know the initial amount \( a = 250 \) mg and we want to find the time \( t \) when the amount decays to \( E(t) = 100 \) mg. Thus, we set up the equation:
\[ 100 = 250 e^{-\frac{t}{500}} \]

3. **Rearrange the equation**:
Divide both sides by 250 to isolate the exponential term:
\[ \frac{100}{250} = e^{-\frac{t}{500}} \]
Simplifying the left side gives:
\[ \frac{2}{5} = e^{-\frac{t}{500}} \]

4. **Take the natural logarithm of both sides**:
To solve for \( t \), we take the natural logarithm (ln) of both sides:
\[ \ln\left(\frac{2}{5}\right) = -\frac{t}{500} \]

5. **Solve for \( t \)**:
Rearranging the equation gives:
\[ t = -500 \ln\left(\frac{2}{5}\right) \]

6. **Calculate \( \ln\left(\frac{2}{5}\right) \)**:
Using a calculator, we find:
\[ \ln\left(\frac{2}{5}\right) \approx -0.9163 \]
Therefore, substituting this value into the equation for \( t \):
\[ t = -500 \times (-0.9163) \approx 458.15 \]

7. **Final answer**:
Rounding to the nearest whole number, it takes approximately:
\[ t \approx 458 \text{ years} \]

### Conclusion:
The time it takes for the initial amount of 250 milligrams to decay to 100 milligrams is approximately **458 years**.