The amount of a sample remaining after [tex]$t$[/tex] days is given by the equation [tex]$P(t)=A(\frac{1}{2})^{\frac{t}{h}}$[/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 sample contains 18% of its original amount of Radon-222. The half-life of Radon-222 is about 3.8 days. Which is the best estimate for the age of the sample?
A. 1.5 days
B. 2.5 days
C. 9.4 days
D. 21.1 days
Answer (1)
Substitute the given values into the equation: 0.18 A = A ( 2 1 ) 3.8 t .
Divide both sides by A : 0.18 = ( 2 1 ) 3.8 t .
Take the natural logarithm of both sides and use the power rule: ln ( 0.18 ) = 3.8 t ln ( 2 1 ) .
Solve for t : t = 3.8 l n ( 0.5 ) l n ( 0.18 ) ≈ 9.4 days. The best estimate for the age of the sample is 9.4 days.
Explanation
Understanding the Problem We are given the formula for the amount of a 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 know that the sample contains 18% of its original amount, so P ( t ) = 0.18 A . The half-life of Radon-222 is h = 3.8 days. We want to find the age of the sample, which is t .
Substituting Values Substitute the given values into the equation: 0.18 A = A ( 2 1 ) 3.8 t .
Simplifying the Equation Divide both sides by A : 0.18 = ( 2 1 ) 3.8 t .
Taking the Logarithm Take the natural logarithm of both sides: ln ( 0.18 ) = ln ( ( 2 1 ) 3.8 t ) .
Applying the Power Rule Use the power rule of logarithms: ln ( 0.18 ) = 3.8 t ln ( 2 1 ) .
Isolating t Solve for t : t = 3.8 ⋅ ln ( 0.5 ) ln ( 0.18 ) .
Calculating t Calculate the value of t : t ≈ 3.8 ⋅ − 0.693147 − 1.714798 ≈ 9.4009 .
Final Answer The age of the sample is approximately 9.4 days.
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 principle is also applied in medicine for radioactive tracers, where the decay rate helps determine how long a substance will remain active in the body.
