1. **Problem statement:** We need to identify which graph represents the solution to the logistic differential equation $$\frac{dP}{dt} = 2P(50 - P)$$ 2. **Understanding the logistic equation:** This equation models population growth with a carrying capacity. The term $50$ is the carrying capacity, meaning the population $P$ will grow and stabilize around $P=50$. 3. **Behavior of solutions:** - When $P$ is small, growth is approximately exponential. - As $P$ approaches $50$, growth slows down. - The population stabilizes at $P=50$ (equilibrium point). 4. **Analyzing the graphs:** - Graph A: Population rises past 50 and continues growing steeply, which contradicts the logistic model since $P$ should not exceed the carrying capacity. - Graph B: Population rises past 50 and continues growing steeply, also contradicting the logistic model. - Graph V: Population starts near zero, grows in an S-shape, and levels off close to 50, matching the logistic model. - Graph G: Population starts near zero, grows in an S-shape, and levels off just above 50, slightly above the carrying capacity, which is unlikely. 5. **Conclusion:** The correct graph is **Graph V**, as it shows the expected logistic growth behavior with carrying capacity at 50. **Final answer:** Graph V