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Home » Combine and add the constant of integration \( C \): - Aurero

Combine and add the constant of integration \( C \): - Aurero

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 The sum of an infinite geometric series is 12, and the first term is 3. Find the common ratio.

📌 Sum formula: \( S = rac{a}{1 - r} \), so \( 12 = rac{3}{1 - r} \).

📌 Solve: \( 12(1 - r) = 3 \) â \( 12 - 12r = 3 \) â \( 12r = 9 \) â \( r = rac{3}{4} \).

📌 A right triangle has legs of lengths \( \sqrt{5} \) and \( \sqrt{11} \). Find the length of the altitude to the hypotenuse.

📌 Hypotenuse: \( c = \sqrt{(\sqrt{5})^2 + (\sqrt{11})^2} = \sqrt{5 + 11} = \sqrt{16} = 4 \).

📌 Area: \( rac{1}{2} \cdot \sqrt{5} \cdot \sqrt{11} = rac{1}{2} \sqrt{55} \).

📌 Also: \( rac{1}{2} \cdot 4 \cdot h = 2h \).

📌 Set equal: \( 2h = rac{1}{2} \sqrt{55} \) â \( h = rac{\sqrt{55}}{4} \).

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