Number of doubling periods: 12 / 3 = 4. Final population = \(150 \cdot 2^4 = 150 \cdot 16 = 2400\).
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- A game developer designs a VR puzzle where a player must rotate a cube to align glowing faces. The cube’s identifier is determined by the least common multiple (LCM) of the numbers 12 and 18, which unlocks the final level. What is the LCM used?
- Prime factorization: 12 = \(2^2 \cdot 3\), 18 = \(2 \cdot 3^2\). LCM = \(2^2 \cdot 3^2 = 4 \cdot 9 = 36\).
- In a virtual ecosystem, the population of digital fireflies doubles every 3 hours. If there are initially 150 fireflies, how many fireflies will exist after 12 hours?
- A biotech startup models bacterial growth in a lab culture with the function \( P(t) = 500 \cdot e^{0.4t} \), where \( t \) is in hours. After how many hours will the population exceed 5000?
- Solve \( 500 \cdot e^{0.4t} > 5000 \) → \( e^{0.4t} > 10 \) → \( 0.4t > \ln(10) \approx 2.3026 \) → \( t > 2.3026 / 0.4 = 5.7565 \).
- Intercept at approximately 5.76 hours; so first integer hour is 6.
