Word Problem- Exponential growth/decay

Question

Suppose that a technician prepares a sample of ^99Tc-pyrophosphate to image the head of a patient suspected of having a heart attack.

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A. At noon, the patient is given 10mici(millicuries) of ^99mTc. If the half life of ^99mTc is 6 hours, write a function of the form Q(t)=Q0(initial)e^-kt to model the radioactivity level Q(t) after t hours.

B. At what time will the level of radioactivity reach 3 miCi?

Answer 1

The basic exponential formula for half life is:

Q(t) = Q₀(1/2)^t/h

Q(t) = amount after t hours

Q₀ = starting amount = 10

t = hours

h = half life period = 6

Q(t) = 10(1/2)^t/6

If you want it in terms of e, (1/2)^t/6 = e^{ln((1/2)^t/6)} = e^(t/6)ln(1/2) = e^-kt where -k = ln(1/2)/6 ≅ -0.1155

Q(t) = 10·e^-0.1155t

For part B, set Q(t) = 3 and solve for t.

3 = 10·e^-0.1155t

3/10 = e^-0.1155t

Take the log (ln) of both sides and solve for t. Use your calculator to compute the answer.