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.

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**.