Cosmic ray bombardment of the atmosphere produces neutrons, which in turn react with nitrogen to produce radioactive carbon-14. Radioactive carbon-14 enters all living tissue through carbon dioxide (via plants). As long as a plant or animal is alive, carbon-14 is maintained in the organism at a constant level. Once the organism dies, however, carbon-14 decays exponentially into carbon-12. By comparing the amount of carbon-14 to the amount of carbon-12 one can determine approximately how long ago the organism died. (Willard Libby won a Nobel Prize for developing this technique for use in dating archaeological specimens). The half-life of carbon-14 is about 5730 years. In answering the following questions, assume that the initial quantity of carbon-14 is 500 milligrams.
a. Construct an exponential function that describles the relationship between A the amount of carbon-14 in milligrams, and t the number of 5730-year time periods.
b. From your table, estimate how many milligrams are left after 25,000 years and after 40,000 years.
Round your answers to two decimal places.
_________milligrams when the number of years is 25,000
_________milligrams when the number of years is 40,000
c.Now construct an exponential function that describes the relationship between A and T where T is measured in years. What is the annual decay factor? The annual decay rate?
Round a to six decimal places.
The exponential function is A=Ca^T where C=____ and A=____
d. Use your function in part (d) to calculate the number of milligrams that would be left after 25,000 years and after 40,000 years.
Round your answers to three decimal places.
A=_______milligrams when the number of years is 25,000
A=_______milligrams when the number of years is 40,000
(a) 500(1/2)^t
(b.1) 500(1/2)^(25000/5730)
= 500(1/2)^4.363 = 24.3
see what you can do from here.
a. The exponential function that describes the relationship between A, the amount of carbon-14 in milligrams, and t, the number of 5730-year time periods, can be constructed using the formula:
A = A₀ * (1/2)^(t/n)
where A₀ is the initial quantity of carbon-14 (500 milligrams) and n is the half-life of carbon-14 (5730 years).
b. To estimate the amount of carbon-14 left after 25,000 years and 40,000 years, we can plug in the values into the equation:
For 25,000 years:
A = 500 * (1/2)^(25000/5730)
A ≈ 45.42 milligrams
For 40,000 years:
A = 500 * (1/2)^(40000/5730)
A ≈ 9.03 milligrams
c. The exponential function that describes the relationship between A and T, where T is measured in years, can be constructed using the formula:
A = C * a^T
where C is the initial quantity of carbon-14 (500 milligrams), and a is the annual decay factor.
To find the annual decay factor, we can use the equation: a = (1/2)^(1/n) where n is the half-life of carbon-14 (5730 years).
a = (1/2)^(1/5730)
a ≈ 0.999879
The annual decay rate can be calculated by subtracting the annual decay factor from 1 and multiplying by 100%. So, the annual decay rate is approximately:
Annual Decay Rate = (1 - a) * 100%
Annual Decay Rate ≈ 0.0121%
Therefore, the exponential function is A = 500 * (0.999879)^T.
d. Using the exponential function in part (c), we can calculate the amount of carbon-14 left after 25,000 years and 40,000 years:
For 25,000 years:
A = 500 * (0.999879)^25000
A ≈ 457.62 milligrams
For 40,000 years:
A = 500 * (0.999879)^40000
A ≈ 90.96 milligrams
a. The exponential function that describes the relationship between A (the amount of carbon-14 in milligrams) and t (the number of 5730-year time periods) can be determined using the half-life formula for exponential decay:
A = A0 * (1/2)^(t/n)
Where A0 is the initial quantity of carbon-14 (500 milligrams) and n is the half-life (5730 years).
Replacing A0 with 500 and n with 5730, the equation becomes:
A = 500 * (1/2)^(t/5730)
b. To estimate the amount of carbon-14 left after a certain number of years, you can substitute the given values of t into the equation from part a.
For 25,000 years:
A = 500 * (1/2)^(25000/5730)
A ≈ 32.84 milligrams
For 40,000 years:
A = 500 * (1/2)^(40000/5730)
A ≈ 6.75 milligrams
c. To construct an exponential function that describes the relationship between A and T (measured in years), we can use the formula:
A = C * a^(T/t)
Where C is the initial quantity of carbon-14 (500 milligrams), a is the annual decay factor, and T is the number of years.
Since the half-life of carbon-14 is 5730 years, the annual decay factor can be calculated as:
a = (1/2)^(1/5730)
The annual decay rate can be calculated by subtracting one from the annual decay factor and multiplying by 100:
Annual decay rate (%) = (a - 1) * 100
Rounding to six decimal places:
a ≈ 0.999123
Annual decay rate ≈ -0.087677%
The exponential function is A = C * a^(T/t), where C is 500 and a is 0.999123.
d. To calculate the number of milligrams that would be left after a certain number of years using the function from part c, substitute the given values of T into the equation.
For 25,000 years:
A = 500 * (0.999123)^(25000/5730)
A ≈ 32.84 milligrams (same result as in part b)
For 40,000 years:
A = 500 * (0.999123)^(40000/5730)
A ≈ 6.75 milligrams (same result as in part b)