Question

Suppose 4 mg of a drug is injected into a person's bloodstream. As the drug is metabolized, the quantity diminishes at the continuous rate of 3% per hour?

Answer 1

Let Q(t) = the mass (mg) remaining after t hours.
We are told that Q diminishes by 3% every hour.

When t = 0, Q = 4 mg
When t = 1, Q = 4*0.97 mg
When t = 2, Q = 4*(0.97)² mg

By induction,

`Q(t) = 4 (0.97)^(t) \, mg`

Q'(t) = 0.97Q
Therefore the rate of decrease is 3% per hour.

The person receives an additional dosage when Q falls to 0.50 mg. This happens when

`4(0.97)^(t)=0.5 \\0.97^(t) = 0.125 \\ t \,ln(0.97) = ln(0.125) \\ t = (ln(0.125))/(ln(0.97)) =68.27 \, hrs`

After the second injection, the mass is now 4.5 mg. Therefore

`Q(t)=4.5(0.97)^(t)`

When the mass again reaches 0.50 mg, then

`4.5(0.97)^(t)=0.5 \\ 0.97^(t)=0.1111 \\ t \, ln(0.97) = ln(0.1111) \\ t = 72.14 \, hrs`

Answers:
(a)
`Q(t) = 4(0.97)^(t)`
(b) 3% per hour
(c) 68.3 hours
(d) 72.1 hours