The half-life of a particular radioactive substance is 1 year. If you started with 220 grams of this substance, how much of it would remain after 7 years?
Remaining Amount $=I(1-r)^t$ grams
Answer (2)
Determine the decay rate r from the half-life: r = 0.5 .
Substitute the initial amount I = 220 , decay rate r = 0.5 , and time t = 7 into the formula: R e mainin g A m o u n t = 220 ( 1 − 0.5 ) 7 .
Calculate ( 0.5 ) 7 = 0.0078125 .
Multiply by the initial amount: 220 × 0.0078125 = 1.71875 . The remaining amount is 1.71875 grams.
Explanation
Understanding the Problem We are given that the half-life of a radioactive substance is 1 year. We start with 220 grams of the substance and want to find out how much remains after 7 years. The formula for the remaining amount is given by R e mainin g A m o u n t = I ( 1 − r ) t , where I is the initial amount, r is the rate of decay, and t is the time in years.
Finding the Decay Rate First, we need to find the decay rate r . Since the half-life is 1 year, after 1 year, half of the substance remains. This means that 1 − r = 0.5 , so r = 0.5 .
Substituting the Values Now, we can substitute the given values into the formula: I = 220 grams, r = 0.5 , and t = 7 years. So, R e mainin g A m o u n t = 220 ( 1 − 0.5 ) 7 = 220 ( 0.5 ) 7 .
Calculating the Power Next, we calculate ( 0.5 ) 7 . The result is 0.0078125 .
Finding the Remaining Amount Finally, we multiply this result by 220: 220 × 0.0078125 = 1.71875 . Therefore, the remaining amount of the substance after 7 years is 1.71875 grams.
Examples
Radioactive decay is used in carbon dating to determine the age of ancient artifacts. By knowing the half-life of carbon-14, scientists can measure the remaining amount of carbon-14 in an artifact and estimate its age. This technique is crucial in archaeology and paleontology for understanding the history of our planet and human civilization. The formula for radioactive decay helps in calculating the age of the artifact based on the amount of radioactive material left.
After 7 years, approximately 1.71875 grams of the radioactive substance will remain from an initial amount of 220 grams, given its half-life is 1 year. This was calculated using the half-life concept and the formula for remaining amount. Each year, the substance reduces to half its amount, which leads to this calculation.
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