Final answer:
To find the amount of radioactive substance remaining after 12 years, we calculate the decay over the additional 3 years as a fraction of the known half-life, which results in approximately 7.143 g of the substance remaining.
Step-by-step explanation:
When dealing with radioactive decay, the amount of a radioactive substance remaining after a certain period can be calculated using the concept of half-life. In this case, since the half-life is 9 years (given that half of the substance decayed from 18 g to 9 g in 9 years), we need to calculate how much will remain after an additional 3 years, making it a total of 12 years.
We can determine the amount remaining after an additional 3 years by realizing that this period constitutes one-third of a half-life. Therefore, the amount does not halve, but decreases according to the exponential decay rule.
Amount remaining after 9 years (one half-life) = 1/2 of the initial amount = 9 g
After 12 years, we can apply the formula: Remaining amount = initial amount * (1/2)^(time elapsed/half-life)
Thus, Remaining amount after 12 years = 9 g * (1/2)^(3/9) = 9 g * (1/2)^(1/3)
To find (1/2)^(1/3), we calculate the cube root of 1/2 which is approximately 0.7937.
Therefore, Remaining amount after 12 years = 9 g * 0.7937 ≈ 7.143 g (rounded to three decimal places)