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Home » For (1,4): \( a + b + c = 4 \) - AIKO, infinite ways to autonomy.

For (1,4): \( a + b + c = 4 \) - AIKO, infinite ways to autonomy.

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 The sum of the first \( n \) natural numbers is given by \( S_n = \frac{n(n+1)}{2} \). What is the sum of the first 50 natural numbers?

📌 Substitute \( n = 50 \) into the formula:

📌 \( S_{50} = \frac{50 \times 51}{2} = 1275 \).

📌 A light beam reflects off a mirror at a 30-degree angle of incidence. What is the angle of reflection?

📌 According to the law of reflection, the angle of reflection is equal to the angle of incidence, which is 30 degrees.

📌 #### 30Question: A satellite dish is shaped like a parabola with a focus at (0, 5). If the dish's opening is 20 units wide, find the equation of the parabola in standard form.

📌 Solution: The standard form of a parabola with vertical axis is $ y = \frac{1}{4p}x^2 $. The focus is at $ (0, p) $, so $ p = 5 $. The width of 20 units implies the parabola passes through $ (10, 5) $. Substituting: $ 5 = \frac{1}{20}(100) \Rightarrow 5 = 5 $, which holds. Thus, the equation is $ y = \frac{1}{20}x^2 $. \boxed{y = \dfrac{1}{20}x^2}

📌 Question: An ichthyologist models fish population growth with a quadratic function $ h(x) $ satisfying $ h(1) = 4 $, $ h(2) = 11 $, and $ h(3) = 22 $. Find $ h(4) $.

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