The concept of Half-Life is not a difficult one to
understand - there is some value X which steadily decreases such that the value repeatedly halves
after some time interval t_h called the "half-life". For example, given t_half = 1 second and
some initial value x_0 = 160, then observe below how the value decreases:
time elapsed | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ... |
value | 160 | 80 | 40 | 20 | 10 | 5 | 2.5 | 1.25 | ... |
This type of pattern has application in real-world fields such as the
Radioactive Decay of isotopes. Take for example
the element Iodine, which typically has 53 protons and
76 neutrons and has a half-life of around 16 million years. If you have 1 gram of Iodine
today, then in 16 million years from now you will only have 0.5 grams of Iodine (the rest
decaying into Xenon, in this case).
However, that's only the case for Iodine atoms having 76 neutrons, called "Iodine-129". Sometimes
Iodine atoms can have 78 neutrons, called "Iodine-131", and Iodine-131 atoms are very unstable,
having a half-life of only 8 days!
But, what about if we wanted to calculate a value between half-life intervals? For example, at 5.5 days? We would encourage you to derive the full equation yourself, but for the sake of getting on with the problem we'll provide it here:
$$\Huge X_t = X_0 \cdot 2^{-({t}/{t_h})}$$
where X_t is the value at the end of the time interval, X_0 is the value at the beginning of
the time interval, t is the time elapsed, and t_h is the half-life.
Another process which behaves similarly is drug concentration in the body, where some quantity of ingested drug enters the bloodstream and is then gradually expelled (usually by "waste") following a half-life dynamic.
John has just returned from the doctor, who perscribed him a brand new medication and instructed John to take one tablet each day. The doctor also told John that its critical he takes the medication regularly as it needs to slowly build up in his system before it becomes effective.
Input Data
First line is Q, the quantity of testcases.
Q lines then follow, each in the format D H A.
D is the Dose of the tablet in mgH is the Half-life of the medication in the human body in hoursA is the Amount of medication that must be present in John's body before it becomes effective, in mgAnswer
Should consist of Q space-separated integers, corresponding to the day at which the amount of
medication in John's body first reaches A.
Assume that John takes his first dose on day 1, and takes each subsequent dose exactly 1 day apart.
Also assume that the entire dose immediately enters John's bloodstream upon ingestion.
Example
input data:
2
1 1 1
123 456 789
answer:
1 8