Question

# In nuclear medicine, a radioactive isotope is used to diagnose certain diseases. A particular isotope has a half-life of 6 hours, which means that every 6 hours, the amount of radioactive isotope is reduced by half due to radioactive decay. In one procedure, 2 milligrams $mg$ of the isotope is injected into the patient. Determine: a) Express the remaining amount of the radioactive isotope as a function of time using a logarithmic exponential function. b) Determine the amount of isotope that will remain in the patient's body after 24 hours.

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## Answer to a math question In nuclear medicine, a radioactive isotope is used to diagnose certain diseases. A particular isotope has a half-life of 6 hours, which means that every 6 hours, the amount of radioactive isotope is reduced by half due to radioactive decay. In one procedure, 2 milligrams $mg$ of the isotope is injected into the patient. Determine: a) Express the remaining amount of the radioactive isotope as a function of time using a logarithmic exponential function. b) Determine the amount of isotope that will remain in the patient's body after 24 hours.

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4.6
**Step a:**
Given exponential decay function:
A$t$ = A_0 \times \left$\frac{1}{2}\right$^{\frac{t}{h}}

For the isotope with a half-life of 6 hours:
A$t$ = 2 \times \left$\frac{1}{2}\right$^{\frac{t}{6}}
A$t$ = 2 \times 2^{-\frac{t}{6}}

**Step b:**
To find the amount of the isotope remaining after 24 hours:
A$24$ = 2 \times 2^{-\frac{24}{6}}
A$24$ = 2 \times 2^{-4}
A$24$ = 2 \times \frac{1}{16}
A$24$ = \frac{2}{16}
A$24$ = \frac{1}{8}

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