Final answer:
To determine the decay rate after 3 hours, the remaining mass after 5 hours, and the initial mass, we can apply the percentage decay rate of 1.5% per hour. After 3 hours, the decay rate is 4.43%, leaving 19.114 mg, and after 5 hours, the remaining mass is 18.55 mg. The initial mass provided in the question is 20 mg.
Step-by-step explanation:
Calculating Radioactive Decay
A 20 mg sample of a radioactive substance decays at a constant rate of 1.5% per hour. To address the questions, we can use the formula for exponential decay, which is:
N(t) = N_0 e^{(kt)}
where N(t) is the remaining amount of substance after time t, N_0 is the initial amount, e is the base of the natural logarithm, and k is the decay constant. However, since we have a percentage decrease per hour, we can also use a simpler approach:
a) The decay rate after 3 hours is calculated by repeatedly applying the 1.5% decay to the substance:
After 1 hour: 20 mg - 1.5% of 20 mg = 19.7 mg
After 2 hours: 19.7 mg - 1.5% of 19.7 mg = 19.405 mg
After 3 hours: 19.405 mg - 1.5% of 19.405 mg = 19.114 mg
Overall decay rate after 3 hours = (20 mg - 19.114 mg) / 20 mg = 4.43%
b) The remaining mass after 5 hours is:
After 4 hours: 19.114 mg - 1.5% of 19.114 mg = 18.827 mg
After 5 hours: 18.827 mg - 1.5% of 18.827 mg = 18.545 mg
So, the remaining mass is 18.55 mg.
c) As there is a constant rate of decay, the initial mass of the substance was 20 mg, as given in the question.