Differential Equations
Exponential decay ODE; amount after time
Grade Class 12
Question:
Ajay takes 100 mg paracetamol every 12 hours. Amount halved every 5 hours. Amount in bloodstream 15 hours after first dose is (mg)
$100\left(\frac12\right)^3$
$100\left[\left(\frac12\right)^3+1\right]$
$100\left[\left(\frac12\right)^3+\left(\frac12\right)^{12/5}\right]$
$100\left[\left(\frac12\right)^3+\left(\frac12\right)^{3/5}\right]$
Step-by-Step Solution
Key Concept: After first dose, amount $=100e^{-kt}$ with $k=\ln2/5$. At $t=12$: amount$=100(1/2)^{12/5}$. At $t=12^+$: add 100 mg. At $t=15$: from first dose: $100(1/2)^3$; from second dose (at $t=12$): $100(1/2)^{3/5}$ (since $15-12=3$ hrs). Total$=100[(1/2)^3+(1/2)^{3/5}]$.
$100\left[\left(\frac12\right)^3+\left(\frac12\right)^{3/5}\right]$.
Correct Answer: 4