Final answer:
The annual rate of decay for a radioactive element with a half-life of 5 years is approximately 13%, calculated using the exponential decay formula and understanding that half of the substance remains after one half-life period.
Step-by-step explanation:
The half-life of a radioactive element is the time it takes for half of the radioactive atoms in a sample to decay into their decay products. For a radioactive element with a half-life of 5 years, this means that after 5 years, only 50% of the original amount would remain. To find the annual rate of decay, we need to understand that the decay follows an exponential decay model rather than a simple linear decay.
To calculate the annual rate of decay, we can use the formula for exponential decay: N(t) = N0 * (1/2)^(t/half-life), where N(t) is the amount remaining after t years, N0 is the initial amount, and the half-life is the half-life of the substance in years. In this case, since we know that after one year, the substance hasn't decayed to half its original amount yet, we look instead at the end of the half-life period. After 5 years, we have 50% of the substance left.
To find the equivalent annual rate, we take the 5th root of 0.5 (since there are 5 years in the half-life period), which gives us approximately 0.87. Subtracting this from 1 and then multiplying by 100 gives us the percentage rate of annual decay. So, the annual rate of decay is approximately 13% rather than 3%, 87%, or 97%.