A drug that halves
Building an exponential model from a half-life and then reading a threshold and a second dose off it.
Scenario
A drug given as a single injection reaches a concentration of 80 mg per litre in the blood, and the concentration halves every 6 hours thereafter.
Assume the halving continues at the same rate for as long as the drug is present.
Answers in order, explanations withheld, and a report on where the chain broke.
- 14.1Exponential growth and decayFoundation
What is the concentration 12 hours after the injection?
- A20 mg per litre
- B40 mg per litre
- C68 mg per litre
- D10 mg per litre
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20 mg per litre
12 hours is two halvings: 80 becomes 40 after 6 hours and 20 after 12. Halving once rather than twice gives 40, which is the answer to a different question.
- 14.2Exponential growth and decayStandard
Builds on: Question 1 supplies the check. Any candidate model that does not return 20 at t = 12 can be discarded without further work.
Which function gives the concentration C, in mg per litre, t hours after the injection?
- AC(t) = 80(1/2)^(t/6)
- BC(t) = 80(1/2)^(6t)
- CC(t) = 80 - (t/6)
- DC(t) = 80(1/6)^(t/2)
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C(t) = 80(1/2)^(t/6)
The base is the factor per period, 1/2, and the exponent counts periods, so it is t divided by the 6-hour period. Multiplying by 6 instead would halve the concentration six times an hour. Test it against question 1: 80(1/2)^(12/6) = 80 x 1/4 = 20.
- 14.3Exponential growth and decayChallenge
Builds on: The model from question 2, though the halving sequence answers it faster than substitution does.
At which of these times is the concentration first below 5 mg per litre?
- A18 hours
- B24 hours
- C30 hours
- D36 hours
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30 hours
The sequence runs 80, 40, 20, 10, 5, 2.5 at 0, 6, 12, 18, 24 and 30 hours. At 24 hours it is exactly 5, which is not below 5, so the first listed time under the threshold is 30 hours. Reading "falls to 5" as "falls below 5" is the whole difficulty here.
- 14.4Interpreting modelsChallenge
Builds on: The 20 mg per litre from question 1. This is the point of the set: the residue of the first dose is only visible if the decay was tracked, and it is what makes repeat dosing accumulate.
A second injection is given 12 hours after the first and raises the concentration by 80 mg per litre. What is the concentration immediately afterwards?
- A100 mg per litre
- B80 mg per litre
- C160 mg per litre
- D20 mg per litre
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100 mg per litre
The second dose adds to what is left rather than replacing it: 20 from the first injection plus 80 gives 100 mg per litre. Answering 80 assumes the blood was clear, and 160 doubles a dose that was never doubled.