Question

The half-life of a certain radioactive material is 42 days. An initial amount of the material has a mass of 511 kg. Find how much radioactive material remains after 6 days. Round to the nearest thousandth.

Answer 1

To determine the amount of radioactive material remaining after 6 days, we can use the formula for exponential decay based on half-life:

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

Where:
- \( N(t) \) is the amount of material remaining after time \( t \),
- \( N_0 \) is the initial amount of material,
- \( t \) is the time elapsed,
- \( t_{1/2} \) is the half-life of the material.

For this problem:
- \( N_0 = 511 \) kg (initial mass of the material),
- \( t = 6 \) days (time elapsed),
- \( t_{1/2} = 42 \) days (half-life of the material).

Next, we can substitute these values into the formula:

\[
N(6) = 511 \left( \frac{1}{2} \right)^{\frac{6}{42}}
\]

First, we simplify \( \frac{6}{42} \):

\[
\frac{6}{42} = \frac{1}{7} \approx 0.142857
\]

Now substitute this value back into the equation:

\[
N(6) = 511 \left( \frac{1}{2} \right)^{0.142857}
\]

To compute \( \left( \frac{1}{2} \right)^{0.142857} \):

\[
\left( \frac{1}{2} \right)^{0.142857} \approx 0.840896
\]

Now calculate \( N(6) \):

\[
N(6) = 511 \times 0.840896 \approx 429.287856
\]

Rounding to the nearest thousandth:

\[
N(6) \approx 429.288 \text{ kg}
\]

Thus, the amount of radioactive material remaining after 6 days is approximately **429.288 kg**.