Drug Decay 68143D
1. **Problem statement:** The amount of a medicinal drug in the body $t$ hours after injection is given by $$D(t) = 23(0.85)^t, \quad t \geq 0,$$ with zero amount before injection.
2. **(a.i) Initial dose:** The initial dose is the amount at $t=0$. Substitute $t=0$ into the formula:
$$D(0) = 23(0.85)^0 = 23 \times 1 = 23$$ milligrams.
3. **(a.ii) Percentage of drug leaving each hour:** The drug amount decreases by a factor of $0.85$ each hour, meaning $85\%$ remains. The percentage leaving is:
$$100\% - 85\% = 15\%$$ per hour.
4. **(b) Amount remaining after 10 hours:** Substitute $t=10$:
$$D(10) = 23(0.85)^{10}.$$ Calculate:
$$0.85^{10} \approx 0.1969,$$
so
$$D(10) \approx 23 \times 0.1969 = 4.53$$ milligrams (rounded to two decimal places).
**Final answers:**
- Initial dose: $23$ mg
- Percentage leaving per hour: $15\%$
- Amount after 10 hours: approximately $4.53$ mg