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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**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}

**Answer:**

After 24 hours,0.125 mg of the isotope would remain in the patient's body.

Given exponential decay function:

For the isotope with a half-life of 6 hours:

**Step b:**

To find the amount of the isotope remaining after 24 hours:

**Answer:**

After 24 hours,

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