How long until the dose drops below 1 mg?

Question

A drug's concentration halves every 4 hours (its half-life). You start with 200 mg in the bloodstream. Solve in your head. How long until less than 1 mg remains? Aim: under 30 seconds.

Accepted Answer

Each half-life divides the amount by 2, so after k half-lives you have \frac{200}{2^{k}} mg. You want this below 1 mg, i.e. 2^{k} 200. Count powers of 2: - 2^{7} = 128 < 200, so after 7 half-lives you still have \frac{200}{128} \approx 1.56 mg, not yet below 1. - 2^{8} = 256 200, so after 8 half-lives you have \frac{200}{256} \approx 0.78 mg, below 1. You need 8 half-lives, and each is 4 hours: 8 	imes 4 = 32 	ext{ hours}. The same 2^{10} \approx 10^{3} toolkit works downward: knocking a quantity down by a factor of 1000 takes about 10 halvings, so clearing a factor of 200 (a bit under 2^8)…

How long until the dose drops below 1 mg?

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