You have 80 grams of a radioactive kind of tellurium. How much will be left after 8 months if its half-life is 2 months?
Understand the Problem
The question is asking for the remaining amount of a radioactive substance after a specific period, given its initial quantity and half-life. To solve this, we will apply the concept of half-life to determine how many half-lives fit into the 8 months and calculate the remaining grams accordingly.
Answer
The remaining amount is $5 \text{ grams}$.
Answer for screen readers
The remaining amount of tellurium after 8 months is $5 \text{ grams}$.
Steps to Solve
- Determine the number of half-lives The first step is to find out how many half-lives fit into 8 months. Given that the half-life is 2 months, we can calculate this by dividing the total time by the half-life.
[ \text{Number of half-lives} = \frac{8 \text{ months}}{2 \text{ months}} = 4 ]
- Calculate the remaining amount after each half-life Next, we start with the initial amount and reduce it by half for each half-life. The formula for the remaining amount after ( n ) half-lives is:
[ \text{Remaining Amount} = \text{Initial Amount} \times \left(\frac{1}{2}\right)^n ]
Substituting the values:
[ \text{Remaining Amount} = 80 \text{ grams} \times \left(\frac{1}{2}\right)^4 ]
- Calculate the final amount Now we compute the remaining amount:
[ \text{Remaining Amount} = 80 \text{ grams} \times \frac{1}{16} = 5 \text{ grams} ]
The remaining amount of tellurium after 8 months is $5 \text{ grams}$.
More Information
After four half-lives (2 months each), the substance has drastically reduced in quantity, illustrating the exponential decay that characterizes radioactive materials.
Tips
- Confusing half-life with total decay time: It's important to remember that the half-life defines the time it takes for half of the substance to decay, not the total time of decay.
- Not performing the calculations correctly for each half-life can lead to incorrect amounts. Always ensure that each calculation is correctly halved as you progress through the half-lives.
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