Answer :
Let's solve the problem step-by-step mathematically to determine how many grams of radon remain from an initial 60 grams after 4 weeks, given that the half-life of radon is about 4 days.
### Step 1: Determine the Total Time in Days
First, we need to convert the time from weeks to days.
- 1 week = 7 days
- 4 weeks = 4 * 7 = 28 days
### Step 2: Calculate the Number of Half-Lives
Next, we calculate how many half-lives have passed in the 28-day period.
- Half-life of radon = 4 days
- Number of half-lives = Total time in days / Half-life
[tex]\[ \text{Number of half-lives} = \frac{28\ \text{days}}{4\ \text{days/half-life}} = 7\ \text{half-lives} \][/tex]
### Step 3: Use the Half-Life Formula to Find the Remaining Mass
To find the remaining mass after a certain number of half-lives, we use the formula:
[tex]\[ \text{Remaining mass} = \text{Initial mass} \times \left( \frac{1}{2} \right)^{\text{Number of half-lives}} \][/tex]
Here,
- Initial mass = 60 grams
- Number of half-lives = 7
Substitute these values into the formula:
[tex]\[ \text{Remaining mass} = 60 \ \text{grams} \times \left( \frac{1}{2} \right)^7 \][/tex]
### Step 4: Calculate the Exponent
[tex]\[ \left( \frac{1}{2} \right)^7 = \frac{1}{2^7} = \frac{1}{128} \][/tex]
### Step 5: Find the Remaining Mass
Now, multiply the initial mass by the result of the exponentiation:
[tex]\[ \text{Remaining mass} = 60 \ \text{grams} \times \frac{1}{128} \][/tex]
### Step 6: Perform the Final Calculation
[tex]\[ \text{Remaining mass} = \frac{60}{128} \][/tex]
To simplify the fraction:
[tex]\[ \text{Remaining mass} = \frac{60}{128} \approx 0.46875 \ \text{grams} \][/tex]
So, after 4 weeks, approximately 0.46875 grams of radon remain from the initial 60 grams.
### Step 1: Determine the Total Time in Days
First, we need to convert the time from weeks to days.
- 1 week = 7 days
- 4 weeks = 4 * 7 = 28 days
### Step 2: Calculate the Number of Half-Lives
Next, we calculate how many half-lives have passed in the 28-day period.
- Half-life of radon = 4 days
- Number of half-lives = Total time in days / Half-life
[tex]\[ \text{Number of half-lives} = \frac{28\ \text{days}}{4\ \text{days/half-life}} = 7\ \text{half-lives} \][/tex]
### Step 3: Use the Half-Life Formula to Find the Remaining Mass
To find the remaining mass after a certain number of half-lives, we use the formula:
[tex]\[ \text{Remaining mass} = \text{Initial mass} \times \left( \frac{1}{2} \right)^{\text{Number of half-lives}} \][/tex]
Here,
- Initial mass = 60 grams
- Number of half-lives = 7
Substitute these values into the formula:
[tex]\[ \text{Remaining mass} = 60 \ \text{grams} \times \left( \frac{1}{2} \right)^7 \][/tex]
### Step 4: Calculate the Exponent
[tex]\[ \left( \frac{1}{2} \right)^7 = \frac{1}{2^7} = \frac{1}{128} \][/tex]
### Step 5: Find the Remaining Mass
Now, multiply the initial mass by the result of the exponentiation:
[tex]\[ \text{Remaining mass} = 60 \ \text{grams} \times \frac{1}{128} \][/tex]
### Step 6: Perform the Final Calculation
[tex]\[ \text{Remaining mass} = \frac{60}{128} \][/tex]
To simplify the fraction:
[tex]\[ \text{Remaining mass} = \frac{60}{128} \approx 0.46875 \ \text{grams} \][/tex]
So, after 4 weeks, approximately 0.46875 grams of radon remain from the initial 60 grams.
