Final answer:
The initial mass of the material is 400 grams. The decay rate is 0.02 per unit of time. After 50 years, approximately 146.86 grams of the material remain. It will take approximately 138.63 years for 100 grams of the material to remain. The half-life of the material is approximately 34.65 years.
Step-by-step explanation:
The given equation represents the decay of a material with time. The equation is m(t) = 400e^(-0.02t), where m(t) is the mass of the material at time t.
To find the initial mass of the material, we can substitute t = 0 into the equation. m(0) = 400e^0 = 400 grams.
The decay rate is given by the exponent in the equation, which is -0.02. This means that the mass of the material decreases at a rate of 0.02 per unit of time.
To find the mass remaining after 50 years, we can substitute t = 50 into the equation. m(50) = 400e^(-0.02*50) = 400e^(-1) ≈ 146.86 grams.
To find the time when there will be 100 grams remaining, we can set m(t) = 100 in the equation and solve for t. 100 = 400e^(-0.02t). Dividing by 400, we get e^(-0.02t) = 0.25. Taking the natural logarithm of both sides, we get -0.02t = ln(0.25). Solving for t, we find t ≈ 138.63 years.
The half-life of a radioactive material is the time it takes for half of the material to decay. In this case, the initial mass is 400 grams and the remaining mass is 200 grams (half of the initial mass). We can set m(t) = 200 in the equation and solve for t to find the half-life. 200 = 400e^(-0.02t). Dividing by 400, we get e^(-0.02t) = 0.5. Taking the natural logarithm of both sides, we get -0.02t = ln(0.5). Solving for t, we find t ≈ 34.65 years.