Cosmic ray bombardment of the atmosphere produces neutrons, which in turn react w/ nitrogen to produce radioactive carbon-14. Radioactive carbon-14 enters all living tissue through carbon dioxide. As long as it 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. The half-life of carbon-14 is about 5730 years. Assume that the initial quantity of carbon-14 is 600 milligrams.

Question

Cosmic ray bombardment of the atmosphere produces neutrons, which in turn react w/ nitrogen to produce radioactive carbon-14. Radioactive carbon-14 enters all living tissue through carbon dioxide. As long as it 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. The half-life of carbon-14 is about 5730 years. Assume that the initial quantity of carbon-14 is 600 milligrams.

the exponential eqution is A=(t)600*(.5)^(t)

BUT

construct an exponential function describing the relationship between A & T where T is measured in years.

Round a to six decimal places.

The exponential function is A=C(a)^T where
C=____ and a=____

Answer 1

A(t)=600*(.5)^(t)

this function gives a half-life of one year. As t grows by 1, the remaining amount is multiplied by 1/2.

So, you want to adjust it so that the exponent grows by 1 as t changes by 5730.

Take a look at the section in your text...

Answer 2

To construct an exponential function describing the relationship between A and T, we need to find the values for C and a.

In this case, we are given that the initial quantity of carbon-14 is 600 milligrams. This means that when T = 0, A = 600.

The exponential equation A = C(a)^T can be rearranged to solve for C and a:

A = 600 = C(a)^0 = C
Therefore, C = 600.

Now we need to find the value of a, which represents the decay factor. The half-life of carbon-14 is about 5730 years. This means that after 5730 years, half of the carbon-14 has decayed, reducing the quantity to 300 milligrams.

Using the formula A = C(a)^T, we can substitute A = 300 and T = 5730 to solve for a:

300 = 600(a)^5730
Divide both sides of the equation by 600:
0.5 = a^5730
Taking the logarithm of both sides (base 10 is usually used in these calculations):
log(0.5) = log(a^5730)
5730*log(a) = log(0.5)
Divide both sides of the equation by 5730:
log(a) = log(0.5) / 5730
Solve for a by taking the antilog of both sides:
a = 10^(log(0.5) / 5730)

Now you can substitute the value of a back into the exponential function A = C(a)^T, with C = 600, to get the final exponential function.