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 sample contains 60% of its original amount of Fermium-257. The half-life of Fermium-257 is about 100 days. About how old is the sample?
A. 52 days
B. 60 days
C. 74 days
D. 136 days
Answer (1)
Substitute the given values into the equation: 0.6 A = A ( 2 1 ) 100 t .
Divide both sides by A: 0.6 = ( 2 1 ) 100 t .
Take the natural logarithm of both sides: ln ( 0.6 ) = 100 t ln ( 2 1 ) .
Solve for t: t = l n ( 0.5 ) 100 l n ( 0.6 ) ≈ 74 .
The age of the sample is approximately 74 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 60% of its original amount, so P ( t ) = 0.6 A . The half-life of Fermium-257 is 100 days, so h = 100 . We need to find the age of the sample, t .
Substituting Values Substitute the given values into the equation: 0.6 A = A ( 2 1 ) 100 t .
Simplifying the Equation Divide both sides by A : 0.6 = ( 2 1 ) 100 t .
Applying Logarithms Take the natural logarithm of both sides: ln ( 0.6 ) = ln ( ( 2 1 ) 100 t ) .
Using Logarithm Properties Use the power rule of logarithms: ln ( 0.6 ) = 100 t ln ( 2 1 ) .
Isolating t Solve for t : t = l n ( 0.5 ) 100 l n ( 0.6 ) .
Calculating t Calculate the value of t . We find that t ≈ 73.69655941662063 . Rounding to the nearest whole number, we get t ≈ 74 days.
Final Answer Therefore, the age of the sample is approximately 74 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 (approximately 5,730 years), scientists can estimate how old the artifact is. This technique is based on the same principles as the problem we solved, where we calculated the age of a Fermium-257 sample based on its remaining amount and half-life. Understanding radioactive decay helps us understand the age and origins of materials around us.
