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Answer :
To solve this problem, we're looking at the radioactive decay of an element with a known half-life. The half-life is the time required for a quantity to reduce to half its initial amount. Let's break down the steps to find out how long it would take for 300 grams of Element X to decay to 80 grams.
Step 1: Understand the formula
The formula for radioactive decay is:
[tex]\[ y = a \times (0.5)^{\frac{t}{\text{half-life}}} \][/tex]
Where:
- [tex]\( y \)[/tex] is the remaining amount of the substance.
- [tex]\( a \)[/tex] is the initial amount.
- [tex]\( t \)[/tex] is the time in minutes.
- The half-life is given as 11 minutes.
Step 2: Plug in the known values
We start with 300 grams and want to find the time for it to decay to 80 grams. Plug these values into the formula:
[tex]\[ 80 = 300 \times (0.5)^{\frac{t}{11}} \][/tex]
Step 3: Solve the equation for [tex]\( t \)[/tex]
To find [tex]\( t \)[/tex], follow these steps:
1. Divide both sides by 300:
[tex]\[ \frac{80}{300} = (0.5)^{\frac{t}{11}} \][/tex]
2. Simplify the fraction:
[tex]\[ \frac{80}{300} = \frac{4}{15} \][/tex]
3. Take the natural logarithm of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ \ln \left(\frac{4}{15}\right) = \frac{t}{11} \times \ln(0.5) \][/tex]
4. Rearrange to isolate [tex]\( t \)[/tex]:
[tex]\[ t = \frac{\ln \left(\frac{4}{15}\right)}{\ln(0.5)} \times 11 \][/tex]
Step 4: Solve for [tex]\( t \)[/tex]
After calculating the natural logarithms and performing the arithmetic:
[tex]\[ t \approx 21.0 \][/tex]
Conclusion:
To the nearest tenth of a minute, it would take approximately 21.0 minutes for 300 grams of Element X to decay to 80 grams.
Step 1: Understand the formula
The formula for radioactive decay is:
[tex]\[ y = a \times (0.5)^{\frac{t}{\text{half-life}}} \][/tex]
Where:
- [tex]\( y \)[/tex] is the remaining amount of the substance.
- [tex]\( a \)[/tex] is the initial amount.
- [tex]\( t \)[/tex] is the time in minutes.
- The half-life is given as 11 minutes.
Step 2: Plug in the known values
We start with 300 grams and want to find the time for it to decay to 80 grams. Plug these values into the formula:
[tex]\[ 80 = 300 \times (0.5)^{\frac{t}{11}} \][/tex]
Step 3: Solve the equation for [tex]\( t \)[/tex]
To find [tex]\( t \)[/tex], follow these steps:
1. Divide both sides by 300:
[tex]\[ \frac{80}{300} = (0.5)^{\frac{t}{11}} \][/tex]
2. Simplify the fraction:
[tex]\[ \frac{80}{300} = \frac{4}{15} \][/tex]
3. Take the natural logarithm of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ \ln \left(\frac{4}{15}\right) = \frac{t}{11} \times \ln(0.5) \][/tex]
4. Rearrange to isolate [tex]\( t \)[/tex]:
[tex]\[ t = \frac{\ln \left(\frac{4}{15}\right)}{\ln(0.5)} \times 11 \][/tex]
Step 4: Solve for [tex]\( t \)[/tex]
After calculating the natural logarithms and performing the arithmetic:
[tex]\[ t \approx 21.0 \][/tex]
Conclusion:
To the nearest tenth of a minute, it would take approximately 21.0 minutes for 300 grams of Element X to decay to 80 grams.
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