The half-life of Palladium-100 is 4 days. After 12 days a sample of Palladium-100 has been reduced to a mass of 5 mg. What was the initial mass (in mg) of the sample? What is the mass (in mg) 6 weeks after the start? You may enter the exact value or round to 4 decimal places.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
The half-life of Palladium-100 is 4 days. After 12 days a sample of Palladium-100 has been reduced to a mass of 5 mg.
What was the initial mass (in mg) of the sample?
What is the mass (in mg) 6 weeks after the start?
![### Example Problem - Half-Life of Palladium-100
**Problem Statement:**
The half-life of Palladium-100 is 4 days. After 12 days, a sample of Palladium-100 has been reduced to a mass of 5 mg.
**Questions to Solve:**
1. **What was the initial mass (in mg) of the sample?**
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2. **What is the mass (in mg) 6 weeks after the start?**
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*Note:* You may enter the exact value or round to 4 decimal places.
### Explanation and Approach:
**Understanding Half-Life:**
The half-life of a radioactive substance is the time it takes for half of the original sample to decay. For Palladium-100, this time period is 4 days.
**Step-by-Step Solution:**
**1. Finding the Initial Mass:**
After 12 days, the mass has reduced to 5 mg. Given that the half-life is 4 days, the initial mass can be calculated as follows:
- 12 days is 3 half-lives (since 12 / 4 = 3).
The decay process can be represented as:
\[ m(t) = m_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \]
Where \( m(t) \) is the mass remaining after time \( t \), \( m_0 \) is the initial mass, \( t \) is the time elapsed, and \( T_{1/2} \) is the half-life.
For \( t = 12 \) days and \( T_{1/2} = 4 \) days:
\[ 5 = m_0 \times \left(\frac{1}{2}\right)^3 \]
\[ 5 = m_0 \times \frac{1}{8} \]
\[ m_0 = 5 \times 8 \]
\[ m_0 = 40 \, \text{mg} \]
So, the initial mass was 40 mg.
**2. Finding the Mass After 6 Weeks:**
First, convert 6 weeks to days:
\[ 6 \, \text{weeks} \times 7 \, \text{days/week} = 42 \, \text{days} \]
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