To determine how much potassium-44 remains in the body after 44 minutes, we can use the concept of half-life. Here is a step-by-step solution:
1. Identify the given data:
- Initial mass of potassium-44 ([tex]\( m_0 \)[/tex]): 7.2 grams
- Half-life of potassium-44 ([tex]\( t_{1/2} \)[/tex]): 22 minutes
- Time elapsed ([tex]\( t \)[/tex]): 44 minutes
2. Calculate the number of half-lives that have passed:
The number of half-lives ([tex]\( n \)[/tex]) is given by:
[tex]\[
n = \frac{t}{t_{1/2}}
\][/tex]
Substituting the given values:
[tex]\[
n = \frac{44 \text{ min}}{22 \text{ min}} = 2
\][/tex]
Hence, 2 half-lives have passed.
3. Calculate the remaining mass:
The remaining mass of a substance after a certain number of half-lives can be determined using the formula:
[tex]\[
m = m_0 \times (0.5)^n
\][/tex]
Substituting the values:
[tex]\[
m = 7.2 \text{ g} \times (0.5)^2
\][/tex]
4. Perform the calculation:
First, calculate [tex]\( (0.5)^2 \)[/tex]:
[tex]\[
(0.5)^2 = 0.25
\][/tex]
Then multiply this by the initial mass:
[tex]\[
m = 7.2 \text{ g} \times 0.25 = 1.8 \text{ g}
\][/tex]
Therefore, the remaining mass of potassium-44 in the body after 44 minutes is 1.8 grams.
