The half-life of a particular radioactive substance is 1 minute. If you started with 375 grams of this substance, how much of it would remain after 5 minutes?
Remaining Amount $= I (1-r)^{ t }$
[?] grams
Round your answer to the nearest whole number.
Answer (1)
Determine the decay rate r from the half-life: r = 0.5 .
Substitute the initial amount I = 375 , decay rate r = 0.5 , and time t = 5 into the formula: R e mainin g A m o u n t = 375 ( 1 − 0.5 ) 5 .
Calculate the remaining amount: 375 \t \t × ( 0.5 ) 5 = 11.71875 .
Round the remaining amount to the nearest whole number: 12 .
Explanation
Understanding the Problem We are given that the half-life of a radioactive substance is 1 minute. We start with 375 grams of the substance and want to find out how much remains after 5 minutes. The formula for the remaining amount is given as R e mainin g A m o u n t = I ( 1 − r ) t , where I is the initial amount, r is the decay rate, and t is the time elapsed.
Finding the Decay Rate First, we need to determine the decay rate, r . Since the half-life is 1 minute, after 1 minute, half of the substance remains. This means that 1/2 = ( 1 − r ) 1 . Solving for r , we get 1 − r = 1/2 , so r = 1/2 = 0.5 .
Identifying Initial Amount and Time Next, we identify the initial amount, I , which is 375 grams, and the time elapsed, t , which is 5 minutes.
Substituting Values into the Formula Now, we substitute I = 375 , r = 0.5 , and t = 5 into the formula: R e mainin g A m o u n t = 375 ( 1 − 0.5 ) 5 = 375 ( 0.5 ) 5 .
Calculating the Remaining Amount We calculate the remaining amount: 375 × ( 0.5 ) 5 = 375 × ( 1/32 ) = 375/32 = 11.71875 .
Rounding to the Nearest Whole Number Finally, we round the remaining amount to the nearest whole number, which is 12.
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 in a sample and estimate how long ago the organism died. This technique is crucial in archaeology and paleontology for understanding the history of life on Earth. For instance, if an artifact initially contained 100 grams of carbon-14 and now contains 25 grams, and given the half-life of carbon-14 is approximately 5,730 years, we can determine that two half-lives have passed, estimating the artifact's age to be around 11,460 years.
