The amount of a sample remaining after [tex]$t$[/tex] days is given by the equation [tex]P(t)=A\left(\frac{1}{2}\right)^{\frac{t}{h}}[/tex], where [tex]$A$[/tex] is the initial amount of the sample and [tex]$h$[/tex] is the half-life, in days, of the substance. A scientist has a [tex]$10-\text{mg}$[/tex] sample of a radioactive isotope. The isotope has a half-life of 8 days. After 16 days, how much of the radioactive isotope remains?
A. 2.0 mg
B. 2.5 mg
C. 5.7 mg
D. 7.1 mg
Answer (2)
Substitute the given values into the formula: P ( 16 ) = 10 ( 2 1 ) 8 16 .
Simplify the exponent: 8 16 = 2 , so P ( 16 ) = 10 ( 2 1 ) 2 .
Evaluate the power: ( 2 1 ) 2 = 4 1 , so P ( 16 ) = 10 ( 4 1 ) .
Calculate the final amount: 10 ( 4 1 ) = 2.5 . The final answer is 2.5 mg .
Explanation
Understanding the Problem We are given the formula for the amount of a radioactive sample remaining after t days: P ( t ) = A ( 2 1 ) h t , where A is the initial amount and h is the half-life. We are given that the initial amount A = 10 mg, the half-life h = 8 days, and we want to find the amount remaining after t = 16 days.
Substituting the Values We substitute the given values into the formula: P ( 16 ) = 10 ( 2 1 ) 8 16 .
Simplifying the Exponent Now we simplify the exponent: 8 16 = 2 , so we have P ( 16 ) = 10 ( 2 1 ) 2 .
Evaluating the Power Next, we evaluate the power: ( 2 1 ) 2 = 4 1 , so we have P ( 16 ) = 10 ( 4 1 ) .
Calculating the Final Amount Finally, we multiply: 10 ( 4 1 ) = 4 10 = 2.5 . Therefore, after 16 days, 2.5 mg of the radioactive isotope remains.
Stating the Answer The amount of the radioactive isotope remaining after 16 days is 2.5 mg.
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. The formula used in this problem is a fundamental tool in these scientific applications.
After using the decay formula, we find that the remaining amount of the radioactive isotope after 16 days is 2.5 mg. This is determined by substituting values into the decay formula and calculating the result step-by-step. The correct answer is option B: 2.5 mg.
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