Final answer:
The mass of the radioactive material remaining after 45 years, originally 195 grams decaying at 4% per year, is computed using an exponential decay formula and rounded to one decimal place, resulting in 32.2 grams.
Step-by-step explanation:
Radioactive Material Decay Calculation
The question involves determining the amount of a radioactive material that remains after a certain period, given its continuous rate of decay. The decay of radioactive materials is an exponential process and is described by decay equations. For a substance that decays at a continuous rate, the amount remaining can be calculated using the formula A(t) = A0e−rt, where A(t) is the amount of substance at time t, A0 is the initial amount, e is the base of natural logarithms, and r is the decay rate.
In this case, the radioactive material has an initial mass of 195 grams and it decays at a rate of 4% per year. To calculate the mass remaining after 45 years, we can substitute the given values into the decay equation:
A(45) = 195 * e−(0.04 * 45).
Performing this calculation:
- A(45) = 195 * e−(1.8)
- A(45) = 195 * 0.16529888822158656
- A(45) = 32.23329717230636 grams
Therefore, after rounding to one decimal place, the mass remaining after 45 years is 32.2 grams.