carbon 14 has a hlf life of 5700 years. scientists use this fact to determine the age of things made up of organic material. Suppose the average page of a book containing approximitly .5 mg og carbon 14 was put into a time capsule. How much carbon14 will each page contain after each of the following of years? a. 5700 b. 11400 c. 22800 d. 34200

If a(t) is the amount of C14 after t years,

a(t) = .05 * 2^(-t/5700)

do you see why this works? every 5700 years, the exponent of t decreases by 1, from 0 to -1 to -2, etc. making the multiplier 1, 1/2, 1/4, ...

a(5700) = .05*2^(-1) = .025
a(11400) = .05*2^(-2) = .0125
and so on

13300

To calculate how much carbon-14 each page of the book will contain after a certain number of years, we need to understand the concept of half-life. The half-life is the time it takes for half of the radioactive substance (in this case, carbon-14) to decay.

Given that the half-life of carbon-14 is 5700 years, we can calculate the amount of carbon-14 remaining after each period of time:

a. After 5700 years:
Since the half-life of carbon-14 is 5700 years, after one half-life (5700 years), half of the initial amount of carbon-14 will remain on each page. Therefore, each page will contain 0.5 mg * 0.5 = 0.25 mg of carbon-14.

b. After 11400 years:
Since another 5700 years have passed, which means two half-lives, we can calculate the amount of carbon-14 remaining on each page. After the first half-life, there is 0.5 mg * 0.5 = 0.25 mg left. After the second half-life, there is 0.25 mg * 0.5 = 0.125 mg left.

c. After 22800 years:
Following the same logic, after three half-lives (3 x 5700 years), we find that each page will have 0.125 mg * 0.5 = 0.0625 mg of carbon-14 left.

d. After 34200 years:
By this point, four half-lives (4 x 5700 years) have passed, so we can calculate the remaining amount of carbon-14 on each page. After one half-life, there is 0.5 mg * 0.5 = 0.25 mg left. After the second half-life, there is 0.25 mg * 0.5 = 0.125 mg left. After the third half-life, there is 0.125 mg * 0.5 = 0.0625 mg left. Finally, after the fourth half-life, there is 0.0625 mg * 0.5 = 0.03125 mg of carbon-14 remaining.

Therefore, after 34200 years, each page of the book will contain approximately 0.03125 mg of carbon-14.