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Home » The total volume of the new solution is \(15 + x\) liters. - inBeat

The total volume of the new solution is \(15 + x\) liters. - inBeat

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Question: What is the smallest positive integer whose square ends in 21?

📌 Question: What is the remainder when the sum of the first 100 positive integers is divided by 17?

📌 This leads to the system of equations:

📌 From (3): $v_2 = -2v_1 + 1$. Substitute into (4):

📌 Complete the square inside each square root:

📌 This represents the sum of distances from the point $(x, 0)$ to the points $(-2, 2)$ and $(2, 2)$ in the plane. The minimal total distance occurs when $(x, 0)$ lies on the reflection of one point across the x-axis and the path is a straight line between the reflected point and the other. Reflecting $(2, 2)$ over the x-axis gives $(2, -2)$. The distance between $(-2, 2)$ and $(2, -2)$ is:

📌 Thus, the minimum value of $f(x)$ is:

📌 Solution: The reflection of a point across the plane $ z = 0 $ simply negates the $ z $-coordinate. Since the bird passes through $ (2, 2, 4) $, its reflection lies at $ (2, 2, -4) $. The parabolas axis being along the $ z $-axis confirms the symmetry, and the reflection is consistent with geometric inversion through the plane.

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