<p>Population doubles in 5 years (\\(\\frac{dP}{dt}=kP\\)). How many years to become 8 times?</p>
Step-by-Step Solution
Key Concept: P = P_0 \cdot e^{kt}. Use doubling to find k.
<div class='solution'><p>$P=P_0e^{kt}$. Doubles in 5 years: $2P_0=P_0e^{5k}$ → $e^{5k}=2$ → $k=\ln2/5$. For 8 times: $8P_0=P_0e^{kt}$ → $e^{kt}=8=2^3$ → $kt=3\ln2$ → $t=3\ln2/k=3\ln2/((\ln2)/5)=15$. <strong>Answer: (2) 15 years</strong>.</p><p class='trap-warning'>⚠️ Trap: Thinking 8=2³ so time = 3×5=15 is correct — but students often just multiply by 4 (to get 20) forgetting that P doubles three times, not four.</p></div>
Correct Answer: 2