Given the half life of a radioactive sample t_1/2 = 245 days. After X days 25% of sample remains. Find X.

The correct answer is 490.

Step-by-step Explanation:

The remaining quantity of a radioactive sample after a certain time is given by the radioactive decay formula:

$N\left(t\right)=N_0{\left(\frac{1}{2}\right)}^{\frac{t}{t_{1/2}}}$

Where:
- $N\left(t\right)$ is the remaining amount of the sample at time $t$.
- $N_0$ is the initial amount of the sample.
- $t_{1/2}$ is the half-life of the sample (given as 245 days).
- $t$ is the elapsed time (equal to $X$ days).

According to the problem, 25% of the sample remains after $X$ days. Therefore, the ratio of the remaining sample to the initial sample is:

$\frac{N\left(X\right)}{N_0}=25\%=0.25=\frac{1}{4}$

Expressing $\frac{1}{4}$ as a power of $\frac{1}{2}$, we get:

$\frac{N\left(X\right)}{N_0}={\left(\frac{1}{2}\right)}^2$

Substituting this into the decay equation:

${\left(\frac{1}{2}\right)}^{\frac{X}{245}}={\left(\frac{1}{2}\right)}^2$

Comparing the exponents on both sides of the equation:

$\frac{X}{245}=2$

Solving for $X$:

$X=2\times 245$

$X=490\text{days}$

Thus, it takes 490 days for 25% of the radioactive sample to remain.