Question

A patient was injected with Tc-99m 24 hrs ago to complete a bone scan and 0.5 mCi remains in her blood stream. If the half life of Tc-99m is 6 hrs. How much Tc-99m was she injected with?

Answer 1

Using the half-life formula for radioactive decay, the patient was originally injected with 8 mCi of Technetium-99m. This calculation was based on the remaining 0.5 mCi in her bloodstream after 24 hours and the half-life of Tc-99m, which is 6 hours.

Step-by-step explanation:

The question involves calculating the initial quantity of a radioactive substance based on its half-life and remaining quantity after a certain period. In this case, the substance is Technetium-99m (Tc-99m), which has a half-life of 6 hours. The patient has 0.5 mCi of Tc-99m remaining in her bloodstream after 24 hours (which equals 4 half-lives, because 24 hrs ÷ 6 hrs/half-life = 4 half-lives).

To find out how much Tc-99m was injected initially, we use the formula:

N(t) = N_0 (1/2)^(t/t_1/2)

Where:

N(t) is the remaining amount of the substance after time t,

N_0 is the initial amount of the substance,

t is the time elapsed,

t_1/2 is the half-life of the substance.

Rearranging the formula to solve for N_0:

N_0 = N(t) / (1/2)^(t/t_1/2)

Putting the values we know into the formula:

N_0 = 0.5 mCi / (1/2)^(24 hrs / 6 hrs)

N_0 = 0.5 mCi / (1/2)^4

N_0 = 0.5 mCi / (1/16)

N_0 = 0.5 mCi × 16

N_0 = 8 mCi

Therefore, the patient was originally injected with 8 mCi of Tc-99m.

Answer 2

To determine the amount of Tc-99m injected, we can use the formula for radioactive decay and solve for the initial amount. Substituting the given values into the equation, we find that the patient was injected with 8 mCi of Tc-99m.

Step-by-step explanation:

To determine how much Tc-99m was injected, we can use the formula for radioactive decay:

A = A0 * (1/2)(t / t1/2)

Where A is the remaining amount, A0 is the initial amount, t is the time passed, and t1/2 is the half-life. We are given that A = 0.5 mCi, t = 24 hrs, and t1/2 = 6 hrs. Substituting the values into the equation:

0.5 mCi = A0 * (1/2)(24 / 6)

Simplifying the equation:

0.5 mCi = A0 * 2-4

0.5 mCi = A0 * 1/16

Now, we can solve for A0 by multiplying both sides of the equation by 16:

A0 = 0.5 mCi * 16 = 8 mCi

Therefore, the patient was injected with 8 mCi of Tc-99m.