The half-life of a particular radioactive substance is 5 months. If you started with 100 grams of this substance, how much of it would remain after 10 months?
Remaining Amount $= I (1-r)^{ t }$
[?] grams
Round your answer to the nearest whole number.
Answer (2)
The half-life of the substance is 5 months, and we start with 100 grams.
After one half-life (5 months), the remaining amount is 2 1 \tmes 100 = 50 grams.
After two half-lives (10 months), the remaining amount is 2 1 \tmes 50 = 25 grams.
Therefore, after 10 months, 25 grams remain: 25 .
Explanation
Understanding the Problem We are given that the half-life of a radioactive substance is 5 months. This means that every 5 months, the amount of the substance is reduced by half. We start with 100 grams of the substance and want to find out how much remains after 10 months. The formula provided is Remaining Amount = I ( 1 − r ) t , where I is the initial amount, r is the rate of decay, and t is the time.
Determining the Decay Rate First, we need to determine the rate of decay. Since the half-life is 5 months, after 5 months, half of the substance remains. This means that after 5 months, we have 50% of the original amount. We can express this mathematically as: ( 1 − r ) 5 = 0.5 However, we can solve this problem more directly using the concept of half-life. After each half-life period, the amount of substance remaining is halved.
Calculating Remaining Amount Since the time period we are interested in (10 months) is exactly two half-lives (2 * 5 months = 10 months), we can calculate the remaining amount by halving the initial amount twice.
After the first 5 months (one half-life), the remaining amount is: 2 1 × 100 = 50 grams After the next 5 months (another half-life, totaling 10 months), the remaining amount is: 2 1 × 50 = 25 grams
Final Answer Therefore, after 10 months, 25 grams of the radioactive substance would remain.
Examples
Radioactive decay is used in carbon dating to determine the age of ancient artifacts. By measuring the amount of carbon-14 remaining in an artifact and knowing its half-life, scientists can estimate how old the artifact is. This technique is crucial in archaeology and paleontology for understanding the history of our planet and human civilization.
After starting with 100 grams of the radioactive substance, after 10 months (which is two half-lives), 25 grams will remain. Each half-life reduces the amount by half, resulting in 50 grams after the first 5 months and 25 grams after the next 5 months. Thus, the answer is 25 grams.
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