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Home » how to make a furnace in minecraft pc - Leaselab

how to make a furnace in minecraft pc - Leaselab

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 We know $ a + b = 4 $, and

📌 a^2 + b^2 = (a + b)^2 - 2ab = 16 - 2ab = 10 \Rightarrow 2ab = 6 \Rightarrow ab = 3

📌 a^3 + b^3 = 4^3 - 3 \cdot 3 \cdot 4 = 64 - 36 = \boxed{28}

📌 Question: Let $ x, y, z $ be positive real numbers such that $ x + y + z = 1 $. Find the minimum value of $ \frac{1}{x} + \frac{4}{y} + \frac{9}{z} $.

📌 Solution: We apply the **Cauchy-Schwarz inequality** in the Engel form (also known as Titu’s Lemma):

📌 \frac{a_1^2}{b_1} + \frac{a_2^2}{b_2} + \frac{a_3^2}{b_3} \ge \frac{(a_1 + a_2 + a_3)^2}{b_1 + b_2 + b_3}

📌 But our expression is $ \frac{1}{x} + \frac{4}{y} + \frac{9}{z} = \frac{1^2}{x} + \frac{2^2}{y} + \frac{3^2}{z} $

📌 Apply Titu’s Lemma:

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Set this equal to the given area:
rac{\sqrt{3}}{4}s^2 = 25\sqrt{3}
Divide both sides by $ \sqrt{3} $:
rac{1}{4}s^2 = 25 \Rightarrow s^2 = 100 \Rightarrow s = 10 ext{ cm}
The new side length is $ s + 4 = 14 $ cm. Compute the new area:
A_{ ext{new}} = rac{\sqrt{3}}{4}(14)^2 = rac{\sqrt{3}}{4} \cdot 196...
\Delta A = 49\sqrt{3} - 25\sqrt{3} = 24\sqrt{3} ext{ cm}^2
Thus, the area increases by $ oxed{24\sqrt{3}} $ cmÂ².
Question: Find the length of the longest altitude of a triangle with s...
Solution: First, compute the area using Heronâs formula. Let $ a = 7...

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