We will use the formula

A(t) = A(0)*e^(-kt)

where A(0) is the initial amount, k is the annual decay rate, t is the time.

We need to find k if t = 462 years and (1/2)A(0) = A(0)*e^(-k*462).

By solving

(1/2)A(0) = A(0)*e^(-k*462) divide both sides by A(0) we have

e^(-k*462) = 1/2

ln(e^(-k*462)) = ln(1/2)

-k*462 = ln(1/2) divide both sides by (-462)

k = -(1/462)*ln(1/2)

by solving we find

k = 0.0015 or 0.15%

click here to see the step by step solution of the equation

The annual decay rate is 0.15%.