Question

The charcoal from ashes found in a cave gave 7.4 14C counts per gram per minute. Wood from the outer portion of a growing tree gives a comparable count of 15.3. The half-life of 14C is 5700 years.

How old are the ashes?
A) 3245 y
B) 5700 y
C) 5970 y
D) 9220 y
E) 5437 y

Answer 1

The age of the ashes can be calculated using the carbon-14 decay count from the sample and comparing it to a living sample. After applying the radioactive decay formula, the age of the ashes is approximately 5936 years, which is closest to option C) 5970 years.

Step-by-step explanation:

To determine the age of the ashes using carbon-14 dating, we use the counts per minute per gram of carbon (expressed as disintegrations per minute, dpm) from the charcoal and compare it to the dpm from a modern sample. Given that the charcoal from the cave gave 7.4 dpm and wood from a living tree gives approximately 15.3 dpm, we can see that the charcoal has less than half the carbon-14 activity of the living sample, indicating it has gone through at least one half-life.

The half-life of carbon-14 is 5,700 years. Using the formula to calculate the age of a sample in a radioactive decay process, we can find the age as follows:

Age = (Half-life) * (log(N0/N) / log(2))

Where N0 is the disintegration rate of the modern sample, N is the disintegration rate of the ancient sample, and log is the logarithm base 2.

Calculating the age:

The calculated age is approximately 5936 years, making the closest option C) 5970 years.

Answer 2

C) 5970 y

Step-by-step explanation:

Given;

initial amount of wood, N₀ = 15.3 cpm/g

remaining amount of wood (charcoal), N = 7.4 cpm/g

half life of carbon 14, t 1/2 = 5700 years

The age of the ashes can be calculated using the following formula;

`N = N_0((1)/(2))^{(t)/(t_1_/_2) }\\\\((1)/(2))^{(t)/(t_1_/_2) } = (N)/(N_0) \\\\((1)/(2))^{(t)/(t_1_/_2) } = (7.4)/(15.3) \\\\((1)/(2))^{(t)/(t_1_/_2) } = 0.48366\\\\t = t_(1/2) Log(1)/(2) (0.48366)\\\\t = (t_(1/2)ln(0.48366))/(-ln(2)) \\\\t = t_(1/2)(1.0479)\\\\t = 5700(1.0479)\\\\t = 5973 \ years\\\\t = 5970 \ years(nearest \ ten)`

Therefore, the ashes are 5970 years