Final answer:
Using the exponential decay model, the equation is set up to find the time when the masses of substances A and B are equal. After solving the equation, it turns out that the substances will have equal amounts after approximately 36.841 years. Given options do not include this result, indicating a possible error in the question or the choices provided.
Step-by-step explanation:
To determine after how many years the amounts of two radioactive substances will be equal, we use the exponential decay model A(t) = A0e−λt, where A(t) is the amount at time t, A0 is the initial amount, and λ is the decay constant.
Let t be the number of years after which substance A and B will have the same mass. The amount of substance A at time t will be 1000e−0.01t and the amount of substance B will be 3000e−0.04t. Setting these two equations equal to each other, we get:
1000e−0.01t = 3000e−0.04t
Dividing both sides by 1000 and e−0.04t, we obtain:
e0.03t = 3
Taking the natural logarithm of both sides, we find:
ln(e0.03t) = ln(3)
0.03t = ln(3)
t = ln(3) / 0.03
Calculating t, we find that the two substances will have the same amount after approximately 36.841 years. Since this value is not in the given options, it seems there may have been an error in the question or the provided options. Therefore, none of the given options (a) 115.525 years, (b) 123.563 years, (c) 121.872 years, (d) 119.346 years are correct.