Question Radioactive technetium- 99 m is often used in diagnostic medicine because it has relatively short half-life but lasts long enough to get the needed testing done on the patient. If its half-life is 6 hours, how much of the radioactive material from a 0.5 ml injection will be in the body in 24 hours? Write your answer rounded to nearest hundredth.

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To determine how much of the radioactive technetium-99m remains in the body after 24 hours, we can use the formula for exponential decay based on half-lives:

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

Where:
- $$ N(t) $$ is the remaining quantity after time $$ t $$,
- $$ N_0 $$ is the initial quantity,
- $$ T_{1/2} $$ is the half-life,
- $$ t $$ is the total time elapsed.

Given:
- $$ N_0 = 0.5 $$ ml,
- $$ T_{1/2} = 6 $$ hours,
- $$ t = 24 $$ hours.

First, we need to calculate how many half-lives fit into 24 hours:

$$
\text{Number of half-lives} = \frac{t}{T_{1/2}} = \frac{24 \text{ hours}}{6 \text{ hours}} = 4
$$

Now we can substitute this into the decay formula:

$$
N(24) = 0.5 \left( \frac{1}{2} \right)^{4}
$$

Calculating $$ \left( \frac{1}{2} \right)^{4} $$:

$$
\left( \frac{1}{2} \right)^{4} = \frac{1}{16} = 0.0625
$$

Now, substituting back into the equation:

$$
N(24) = 0.5 \times 0.0625 = 0.03125 \text{ ml}
$$

Finally, rounding to the nearest hundredth:

$$
N(24) \approx 0.03 \text{ ml}
$$

Thus, the amount of radioactive technetium-99m remaining in the body after 24 hours is approximately **0.03 ml**.
After 24 hours, approximately 0.03 ml of the radioactive material remains in the body.

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