A community health researcher is examining the growth of urban gardens, where the number of plants in a garden can be modeled by the function \( G(t) = rac{4t^2 + 3}{t - 1} \). Determine the value of \( G(3) \) and simplify the result.
["Revolutionizing Urban Spaces: How Community Gardens Thrive Using Data-Driven Insights", "Urban gardens are becoming a cornerstone of sustainable city living, offering greenery, fresh food, and community connection in densely populated areas. A key aspect of managing these green spaces is understanding plant growth patterns—particularly how the number of plants evolves over time. Recent studies led by community health researchers have introduced a powerful model to predict plant density in urban gardens, helping planners and gardeners optimize space and resources.", "One such model, ( G(t) = \frac{4t^2 + 3}{t - 1} ), relates the number of plants (( G )) to time in months (( t )), where ( t <br/>\neq 1 ) due to a vertical asymptote indicating abrupt changes in growth capacity. Understanding how this function behaves at specific values is essential for effective garden management and resource allocation.", "### Calculating Plant Growth: A Closer Look at ( G(3) )", "To assess growth at ( t = 3 ) months—representing a mid-season evaluation—we compute ( G(3) ):", "[\nG(3) = \frac{4(3)^2 + 3}{3 - 1}\n]", "First, calculate the numerator:", "[\n4(3)^2 + 3 = 4 \ imes 9 + 3 = 36 + 3 = 39\n]", "Next, compute the denominator:", "[\n3 - 1 = 2\n]", "Now divide:", "[\nG(3) = \frac{39}{2} = 19.5\n]", "Thus, at 3 months, the garden contains 19.5 plants on average—representing a hybrid figure indicating mixed planting density, possibly with partial mature and new growth. While fractional plants aren’t tangible, the value reflects an average density suitable for model calibration.", "This precise calculation enables urban researchers to track seasonal changes and adjust garden designs for optimal biodiversity and yield.", "### The Bigger Picture: Modeling Urban Green Growth", "Using functions like ( G(t) ), community health researchers can interpolate growth trends, identify bottlenecks, and evaluate the impact of shared garden programs. The simplicity and predictive power of such models make them invaluable tools in urban planning, fostering healthier, more resilient neighborhoods.", "As cities grow, data-driven approaches to urban green spaces—anchored by clear mathematical models—will help nurture both people and ecosystems in tandem.", "---", "Key Takeaway: Evaluating ( G(3) = 19.5 ) shows how mathematical modeling brings urban gardening into a measurable, analyzable domain—empowering communities to grow smarter, together."]
