A certain radioactive substance has a half-life of 1 hour. If you start with 1 gram of the material at noon, how much will be left at 3:00 P.M.? At 6:00 P.M.? At 10:00 P.M.?
To determine how much of the radioactive substance is left at different times, we need to understand the concept of half-life. The half-life is the time it takes for half of the original substance to decay.
In this case, the half-life is 1 hour. So, after each hour, half of the substance will decay, and only half will remain.
Let's calculate the amount remaining at the given times:
1. From noon (12:00 P.M.) to 3:00 P.M., three hours have passed. Within these three hours, there will be three sets of half-life decays (12:00-1:00 P.M., 1:00-2:00 P.M., and 2:00-3:00 P.M.). Each of these sets reduces the remaining amount by half, so we multiply the initial amount by (1/2) three times:
Remaining amount at 3:00 P.M. = 1 gram * (1/2) * (1/2) * (1/2) = 1 gram * (1/8) = 1/8 gram
Therefore, at 3:00 P.M., there will be 1/8 gram of the radioactive substance left.
2. From noon (12:00 P.M.) to 6:00 P.M., six hours have passed. Within these six hours, there will be six sets of half-life decays. Using the same calculation as above:
Remaining amount at 6:00 P.M. = 1 gram * (1/2) * (1/2) * (1/2) * (1/2) * (1/2) * (1/2) = 1 gram * (1/64) = 1/64 gram
Therefore, at 6:00 P.M., there will be 1/64 gram of the radioactive substance left.
3. From noon (12:00 P.M.) to 10:00 P.M., ten hours have passed. Within these ten hours, there will be ten sets of half-life decays. Applying the same calculation:
Remaining amount at 10:00 P.M. = 1 gram * (1/2)^(10) = 1 gram * (1/1024)
Therefore, at 10:00 P.M., there will be 1/1024 gram of the radioactive substance left.
To summarize:
- At 3:00 P.M., there will be 1/8 gram of the radioactive substance left.
- At 6:00 P.M., there will be 1/64 gram of the radioactive substance left.
- At 10:00 P.M., there will be 1/1024 gram of the radioactive substance left.