An animal dies in the forest. How long will it be until only #7/8# of the original amount of Carbon 14 remains?

Question

Recall that the half life of Carbon 14 is 5568 years. Round to the nearest year.

NOTE: The half-life is incorrect. It is 5730 years.
- Truong-Son

Peyton Adams · Accepted Answer

#1073# years

All thanks to Truong-Son Nguyen, aka his method! We use one of the radioactive decay equation, which states that #[A]=[A]_0e^(-kt)# #[A]# is the amount of substance right now #[A]_0# is the original amount #-k# is the decay constant, and it is negative because the system experiences decay #t# is the time in years We also have half-life, #t_(1/2)#, related by the equation, #t_(1/2)=(ln 2)/k# From the original equation, we have #[A]=7/8[A_0]#, and so: #[A]/[A_0]=7/8=e^(-kt)# #:.kt=-ln(7/8)=0.1335# Therefore, the decay constant becomes, #k=(ln 2)/(t_(1/2))=(0.693)/(5568)=1.24*10^-4 \ "years"^-1# And so, we get, #t=0.1335/k=0.1335/(1.24*10^-4 \ "years"^-1)~~1073# years

Wyatt Atkins · Answer

undefined The half-life of Carbon-14 is approximately 5730 years. Therefore, to determine the time it takes for 7/8 of the original amount of Carbon-14 to remain, we need to find the number of half-lives required. Since 7/8 is equal to 0.875, we need to solve for $ n $ in the equation $ \left(\frac{1}{2}\right)^n = 0.875 $. By taking the logarithm of both sides, we find $ n \approx 3.21 $ half-lives. Multiplying this by the half-life of Carbon-14 (5730 years) yields approximately 18390 years. Therefore, it would take approximately 18390 years for 7/8 of the original amount of Carbon-14 to remain after an animal dies in the forest.

An animal dies in the forest. How long will it be until only #7/8# of the original amount of Carbon 14 remains?

Recall that the half life of Carbon 14 is 5568 years. Round to the nearest year. NOTE: The half-life is incorrect. It is 5730 years. - Truong-Son

Recall that the half life of Carbon 14 is 5568 years. Round to the nearest year.

NOTE: The half-life is incorrect. It is 5730 years.
- Truong-Son