An isotope of technetium is used to prepare images of internal body organs. This isotope has a half-life (time required for half the material to erode) of approximately 6 h. If a patient is injected with 30 mg of this isotope, what will be the technetium level in the patient after 4 h? Use the function
A(t) = 30

1
2
t/6

,
where A is the technetium level, in milligrams, in the patient after t hours. Round to the nearest tenth of a microgram.

To solve this problem, we will use the given formula:

\[
A(t) = 30 \times \left(\frac{1}{2}\right)^{\frac{t}{6}}
\]

Here:
- \( A(t) \) is the amount of technetium remaining after \( t \) hours.
- \( t \) is the time in hours.
- The initial amount of technetium is 30 mg.

We are asked to find the amount of technetium remaining after 4 hours (\( t = 4 \)).

Let's substitute \( t = 4 \) into the equation and solve for \( A(4) \).

\[
A(4) = 30 \times \left(\frac{1}{2}\right)^{\frac{4}{6}}
\]

First, calculate the exponent:

\[
\frac{4}{6} = \frac{2}{3}
\]

Now, calculate \( \left(\frac{1}{2}\right)^{\frac{2}{3}} \) and then multiply by 30. I'll do the calculation next.The amount of technetium remaining in the patient after 4 hours is approximately 18.9 mg.

Would you like to know more details or have any other questions?

Here are some related questions you might find interesting:
1. How much technetium remains after 8 hours?
2. How is the concept of half-life important in medicine?
3. How do you calculate the half-life if it's not given?
4. What is the difference between exponential decay and linear decay?
5. How does the half-life affect the duration of effectiveness in medical treatments?

**Tip:** When dealing with half-life problems, remember that each half-life period reduces the substance by half, which can be modeled using exponential functions.

Explore how to calculate the amount of technetium remaining in a patient's body after 4 hours using exponential decay. Understand the concept of half-life in medicine and its application to technetium isotopes. Get a detailed solution with step-by-step calculations and learn about the exponential decay formula A(t) = 30 * (1/2)^(t/6).

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