Carbon 14, with a half-life of 5700 years, is used to estimate the age of organic materials. What fraction of the original amount of carbon 14 would an object have if it were 2000 years old?
A) $e ^ { - ( 57 / 20 ) \ln 2 }$
B) $\frac { 57 } { 20 } \ln 2$
C) $e ^ { - ( 20 / 57 ) \ln 2 }$
D) $\frac { 20 } { 57 } \ln 2$
E) $e ^ { ( 57 / 20 ) \ln 2 }$
F) $\frac { 1 } { 57 } \ln 20$
G) $e ^ { ( 20 / 57 ) \ln 2 }$
H) $\frac { 1 } { 20 } \ln 57$

Question

Carbon 14, with a half-life of 5700 years, is used to estimate the age of organic materials. What fraction of the original amount of carbon 14 would an object have if it were 2000 years old?
A) $e ^ { - ( 57 / 20 ) \ln 2 }$ 
B) $\frac { 57 } { 20 } \ln 2$ 
C) $e ^ { - ( 20 / 57 ) \ln 2 }$ 
D) $\frac { 20 } { 57 } \ln 2$ 
E) $e ^ { ( 57 / 20 ) \ln 2 }$ 
F) $\frac { 1 } { 57 } \ln 20$ 
G) $e ^ { ( 20 / 57 ) \ln 2 }$ 
H) $\frac { 1 } { 20 } \ln 57$

Quizplus · Accepted Answer

The fraction of the original amount of carbon-14 remaining after time $ t $ is given by $ e^{-kt} $, where $ k = \frac{\ln 2}{\text{half-life}} $. For 2000 years, $ k = \frac{\ln 2}{5700} $, so the fraction is $ e^{-(2000/5700)\ln 2} = e^{-(20/57)\ln 2} $.

Carbon 14, with a Half-Life of 5700 Years, Is Used e−(57/20)ln⁡2e ^ { - ( 57 / 20 ) \ln 2 }e−(57/20)ln2

Carbon 14, with a Half-Life of 5700 Years, Is Used $e ^ { - ( 57 / 20 ) \ln 2 }$