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the westin san antonio north - Redraw

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Question: What is the smallest three-digit number representing the population threshold for a community, divisible by both 12 and 18, ensuring equitable resource distribution?

📌 Solution: Find the LCM of 12 and 18. Prime factors: $12 = 2^2 imes 3$, $18 = 2 imes 3^2$. LCM is $2^2 imes 3^2 = 36$. The smallest three-digit multiple of 36 is $36 imes 3 = 108$. The population threshold is $oxed{108}$.

📌 Question: What is the least common multiple of 15 and 20, representing the synchronized cycles of two medical devices used in patient monitoring?

📌 Solution: Prime factorizations: $15 = 3 imes 5$, $20 = 2^2 imes 5$. LCM is $2^2 imes 3 imes 5 = 60$. The synchronized cycle length is $oxed{60}$.Question: A civil engineer designs a parabolic arch with the equation $ y = -ax^2 + bx + c $. If the arch's highest point is at $ (3, 10) $ and passes through $ (0, 4) $, find the value of $ a $.

📌 Solution: The vertex form of a parabola is $ y = -a(x - h)^2 + k $, where $ (h, k) $ is the vertex. Given the vertex $ (3, 10) $, the equation becomes $ y = -a(x - 3)^2 + 10 $. Substituting the point $ (0, 4) $:

📌 $$ 4 = -a(0 - 3)^2 + 10 \implies 4 = 9a + 10 \implies 9a = -6 \implies a = -rac{2}{3}. $$

📌 However, since the coefficient of $ x^2 $ is given as $ -a $, and parabolic arches open downward, $ a $ must be positive. Correcting the sign:

📌 $$ 4 = -a(9) + 10 \implies -9a = 6 \implies a = -rac{2}{3} \quad ext{(invalid)}. $$

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