Question involving exponential decay

Question

The half-life of radium-223 is approximately 1143 years. Use the formula A(t)=A_0 a^t to determine a and write a general formula, in the form A(t)=A_0 a^t, that describes the amount of radium-223 left after t years, where A_0 is the amount of radium-223 at time t=0. Round a to six decimal places.

Answer 1

The half-life, by definition, is the time it takes for your initial amount of material to decrease to half of its amount. Thus, we can define the following variables:

t_1/2 = 1143 yrs

A(t_1/2) = (A_0)/2

With these, let us plug these into the decay formula:

A(t_1/2) = (A_0)/2 = A_0 * a^t1/2

By eliminating A_0 and substituting in t_1/2:

1/2 = a¹¹⁴³

ln(1/2) = 1143 * ln(a)

a = e^{(ln(1/2)/1143)}

a = 0.999394

Thus, the final equation becomes:

A(t) = A_0 * 0.999394^t