A scientist has 100 milligrams of a radioactive element. The amount of radioactive element remaining after [tex]$t$[/tex] days can be determined using the equation [tex]$f(t)=100\left(\frac{1}{2}\right)^{\frac{t}{10}}$[/tex]. After three days, the scientist receives a second shipment of 100 milligrams of the same element. The equation used to represent the amount of shipment 2 remaining after [tex]$t$[/tex] days is [tex]$f(t)=100\left(\frac{1}{2}\right)^{\frac{t-3}{10}}$[/tex]. After any time, [tex]$t$[/tex], the mass of the element remaining in shipment 1 is what percentage of the mass the element remaining in shipment 2?
A. 78.1 %
B. 81.2 %
C. 123.1 %
D. 128.0%
Answer (1)
Define the mass of shipment 1 as m 1 ( t ) = 100 ( 2 1 ) 10 t and the mass of shipment 2 as m 2 ( t ) = 100 ( 2 1 ) 10 t − 3 .
Calculate the percentage P = m 2 ( t ) m 1 ( t ) × 100 .
Simplify the expression to P = ( 2 1 ) 10 3 × 100 .
Calculate the final percentage: 81.2% .
Explanation
Understanding the Problem We are given two equations that represent the amount of a radioactive element remaining after t days for two shipments. The first shipment's remaining amount is given by f ( t ) = 100 ( 2 1 ) 10 t , and the second shipment's remaining amount is given by f ( t ) = 100 ( 2 1 ) 10 t − 3 . We want to find the percentage of the mass of the first shipment remaining compared to the mass of the second shipment.
Defining Variables and the Percentage Let m 1 ( t ) be the mass of shipment 1 remaining after t days, so m 1 ( t ) = 100 ( 2 1 ) 10 t .
Let m 2 ( t ) be the mass of shipment 2 remaining after t days, so m 2 ( t ) = 100 ( 2 1 ) 10 t − 3 .
We want to find the percentage P = m 2 ( t ) m 1 ( t ) × 100 .
Substituting the Given Equations Substitute the expressions for m 1 ( t ) and m 2 ( t ) into the equation for P: P = 100 ( 2 1 ) 10 t − 3 100 ( 2 1 ) 10 t × 100
Simplifying the Expression Simplify the expression: P = ( 2 1 ) 10 t − 3 ( 2 1 ) 10 t × 100 = ( 2 1 ) 10 t − 10 t − 3 × 100 = ( 2 1 ) 10 3 × 100
Calculating the Percentage Calculate the value of ( 2 1 ) 10 3 × 100 :
( 2 1 ) 10 3 × 100 ≈ 81.225
Final Answer The percentage of the mass of the element remaining in shipment 1 compared to the mass of the element remaining in shipment 2 is approximately 81.225%. Comparing this to the given options, the closest one is 81.2%.
Examples
Radioactive decay is used in various fields, including medicine and archaeology. For example, in medicine, radioactive isotopes are used for diagnostic imaging and cancer treatment. In archaeology, carbon-14 dating is used to determine the age of organic materials. Understanding the percentage of radioactive material remaining after a certain time helps scientists and doctors to accurately measure and apply radioactive substances in these fields. This problem demonstrates how to calculate the relative amounts of radioactive material from two different shipments over time, which is crucial for managing and tracking radioactive substances in practical applications.
