The half-life of a radioactive kind of antimony is 4 days. How much will be left after 12 days, if you start with 7,600 grams of it?
Understand the Problem
The question is asking how much of a radioactive substance will remain after a given time, knowing its half-life and the initial amount. We will solve this by using the concept of half-lives over the specified time period.
Answer
After 12 days, $950$ grams of the substance will remain.
Answer for screen readers
After 12 days, 950 grams of the radioactive substance will remain.
Steps to Solve
-
Determine the number of half-lives Calculate how many half-lives fit into the total time of 12 days. Since the half-life is 4 days, you can find this by dividing the total time by the half-life: $$ \text{number of half-lives} = \frac{12 \text{ days}}{4 \text{ days}} = 3 $$
-
Calculate the remaining amount For each half-life, the remaining amount of substance is halved. Start with the initial amount of 7,600 grams and apply the formula: $$ \text{Remaining amount} = \text{Initial amount} \times \left( \frac{1}{2} \right)^{\text{number of half-lives}} $$ Substituting the values gives: $$ \text{Remaining amount} = 7600 \times \left( \frac{1}{2} \right)^{3} $$
-
Compute the final amount Now calculate the remaining amount: $$ \text{Remaining amount} = 7600 \times \left( \frac{1}{2} \right)^{3} = 7600 \times \frac{1}{8} $$
-
Final calculation Calculating this will give you: $$ \text{Remaining amount} = \frac{7600}{8} = 950 $$
After 12 days, 950 grams of the radioactive substance will remain.
More Information
This problem illustrates the concept of exponential decay, commonly encountered in radioactive decay and other natural processes. The calculation method used—dividing the time period by the half-life—allows us to understand how much of a substance remains over time.
Tips
- Miscalculating half-lives: Make sure to divide the total time by the half-life accurately.
- Forgetting to account for multiple half-lives: After determining how many half-lives have passed, ensure each one is accounted for in the remaining amount calculation.
AI-generated content may contain errors. Please verify critical information
