The half-life of a particular radioactive substance is 1 year. If you started with 50 grams of this substance, how much of it would remain after 3 years?
Remaining Amount $=50$ ( $1-$ [?])
Remaining Amount $= I (1-r)^{ t }$
Answer (1)
Determine the decay rate: Since the half-life is 1 year, the decay rate r = 0.5 .
Apply the exponential decay formula: Remaining Amount = I ( 1 − r ) t , where I = 50 grams and t = 3 years.
Substitute the values: Remaining Amount = 50 ( 0.5 ) 3 .
Calculate the remaining amount: 50 \t × 0.125 = 6.25 grams. The final answer is 6.25 .
Explanation
Understanding the Problem We are given that the half-life of a radioactive substance is 1 year. This means that after each year, the amount of the substance is reduced by half. We start with 50 grams of the substance and want to find out how much remains after 3 years. We are given the formula for the remaining amount: Remaining Amount = I ( 1 − r ) t , where I is the initial amount, r is the decay rate, and t is the time in years.
Finding the Decay Rate First, we need to determine 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 .
Applying the Formula Now we can use the formula for exponential decay: Remaining Amount = I ( 1 − r ) t , where I = 50 grams, r = 0.5 , and t = 3 years.
Substituting the Values Substitute the values into the formula: Remaining Amount = 50 ( 1 − 0.5 ) 3 = 50 ( 0.5 ) 3 .
Calculating the Power Calculate ( 0.5 ) 3 = 0.125 .
Calculating the Remaining Amount Finally, calculate the remaining amount: 50 × 0.125 = 6.25 grams.
Final Answer Therefore, after 3 years, 6.25 grams of the radioactive substance would remain.
Examples
Radioactive decay is used in carbon dating to determine the age of ancient artifacts. For example, if an artifact initially contained 50 grams of carbon-14, which has a half-life of about 5,730 years, we can use the decay formula to estimate how old the artifact is based on the amount of carbon-14 remaining. This technique is invaluable in archaeology and paleontology.
