Question

The half-life of the radioactive isotope Carbon-14 is about 5730 years.

Part (a) - Find N in terms of t.
The amount of a radioactive element A at time t is given by the formula
A(t) = A0ekt,
where A0 is the initial amount of the element and k < 0 is the constant of proportionality which satisfies the equation
(instantaneous rate of change A(t) at time t) = kA(t).

Use the formula given above to express the amount of Carbon-14 left from an initial N milligrams as a function of time t in years. (Round k to six decimal places.)
A(t) = mg

Answer 1

The amount of Carbon-14 remaining after time t can be expressed as A(t) = Ne^{kt}, with the decay constant k calculated from the half-life of 5730 years to be approximately −0.000121.

Step-by-step explanation:

The half-life of Carbon-14 is 5730 years, and this property is used to determine the decay constant. The decay constant (k) is found using the relationship between half-life and decay constant, which is given by k = −(0.693/t_{1/2}). For Carbon-14 with a half-life (t_{1/2}) of 5730 years, the decay constant k would be k = −(0.693/5730) years−1, which simplifies to k = −0.000121 (rounded to six decimal places).

Using the initial amount of Carbon-14 as N milligrams, the amount A(t) remaining after time t in years can be expressed as A(t) = Ne^{kt}. This equation is derived from the exponential decay formula A(t) = A_{0}e^{kt}, where A_{0} is the initial amount, k is the decay constant, and t is the time in years.