A logistic growth model for a population in an urban zone is given by the equation \( P(t) = \frac{1000}{1 + 9e^{-0.5t}} \). Find the \( y \)-intercept of this curve.

["Understanding the Logistic Growth Model for Urban Population Dynamics", "In the field of urban planning and population studies, modeling how populations grow over time is crucial for infrastructure development, resource allocation, and sustainable city growth. One widely used model is the logistic growth model, which accounts for limited resources and environmental constraints—noted by an S-shaped curve that reflects initial exponential growth followed by a slowdown as capacity is approached.", "### Application to Urban Zones", "The logistic model for population ( P(t) ) in an urban zone is given by:\n[\nP(t) = \frac{1000}{1 + 9e^{-0.5t}}\n]\nHere, the carrying capacity ( K = 1000 ) represents the maximum sustainable population of the city zone, limited by space, services, and resources. The parameter ( 0.5 ) governs the growth rate, and the observation time ( t ) is measured in appropriate units (e.g., years).", "### The ( y )-Intercept: Initial Population", "A key feature of this logistic curve is its ( y )-intercept, which occurs at ( t = 0 )—the time when the population is first measured. At this moment, the population ( P(0) ) reveals the starting point of urban growth.", "Substitute ( t = 0 ) into the equation:\n[\nP(0) = \frac{1000}{1 + 9e^{-0.5 \cdot 0}} = \frac{1000}{1 + 9 \cdot 1} = \frac{1000}{1 + 9} = \frac{1000}{10} = 100\n]", "Thus, the ( y )-intercept is at the point ( (0, 100) ), meaning the urban population begins with 100 residents at the start of observation.", "### Why the ( y )-Intercept Matters", "Understanding the initial population is essential for urban planners:\n- It sets the baseline for infrastructure needs.\n- It informs projections for healthcare, housing, and education.\n- Combined with growth dynamics, it allows forecasting approaching saturation points as the city nears its carrying capacity.", "In summary, the logistic model ( P(t) = \frac{1000}{1 + 9e^{-0.5t}} ) effectively captures urban population growth with a clear starting baseline of 100 individuals, illustrated by its ( y )-intercept at ( (0, 100) ).", "---", "Optimizing city development begins with accurate modeling—the logistic equation provides a powerful foundation for understanding and planning urban population dynamics."]