Question:** An anthropologist is studying the relationship between community size and resource allocation. In a hypothetical community, the number of resources \( R \) is directly proportional to the square of the community size \( S \), and inversely proportional to the number of leaders \( L \). If \( R = k \cdot \frac{S^2}{L} \) where \( k \) is a constant, find \( S \) when \( R = 50 \), \( L = 5 \), and \( k = 2 \).

["### How Community Size and Leadership Shape Resource Allocation: A Mathematical Exploration", "Understanding how communities allocate resources is fundamental to anthropology and sociology. One intriguing relationship emerges when studying hypothetical communities: the amount of resources ( R ) available depends on three key factors: community size ( S ), the number of leaders ( L ), and a proportionality constant ( k ). The formula\n[\nR = k \cdot \frac{S^2}{L}\n]\ncaptures this dynamic, showing that resources grow quadratically with community size but shrink inversely with leadership numbers.", "In a real-world application of this model, consider an anthropological study examining a small community with ( R = 50 ) resources, led by ( L = 5 ) leaders, and guided by a constant ( k = 2 ). To assess how community size ( S ) contributes to resource distribution in this context, we solve for ( S ).", "Start with the given equation:\n[\n50 = 2 \cdot \frac{S^2}{5}\n]", "Simplify the right-hand side:\n[\n50 = \frac{2S^2}{5}\n]", "Multiply both sides by 5 to eliminate the denominator:\n[\n250 = 2S^2\n]", "Divide both sides by 2:\n[\n125 = S^2\n]", "Take the square root of both sides:\n[\nS = \sqrt{125} = 5\sqrt{5}\n]", "Thus, the community size ( S ) required to sustain 50 resources under this model is ( 5\sqrt{5} ), approximately 11.18 people. This demonstrates how mathematical modeling illuminates the balance between social structure and resource availability in community planning.", "By analyzing relationships like ( R = k \cdot \frac{S^2}{L} ), anthropologists gain quantitative insights into sustainability, leadership roles, and collective well-being—evidence of how data-driven methods enrich cultural and social analysis."]