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📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 5Question: A spherical protein structure has radius $ 2r $ units, while a hemispherical lipid droplet has radius $ r $ units. What is the ratio of the volume of the protein to the volume of the lipid droplet?

📌 Question: A right triangle has hypotenuse $ h $ and inradius $ r $. What is the ratio of the area of the incircle to the area of the triangle?

📌 Question: In a triangle with side lengths $ 13 $, $ 14 $, and $ 15 $, find the length of the shortest altitude.

📌 Question: A fossilized microorganism forms a regular hexagon inscribed in a circle of radius $ R $. What is the area of the hexagon in terms of $ R $?

📌 $$Question: A volcanologist records seismic activity levels as integers and notices that the cube of a certain activity level ends in 888. Find the smallest positive integer whose cube ends in 888.

📌 We solve this using modular arithmetic, breaking it into modulo 8 and modulo 125, then applying the Chinese Remainder Theorem.

📌 So $ k = 3 + 5m $, $ n = 2 + 5(3 + 5m) = 17 + 25m $.

📌 So $ m = 2 + 5t $, $ n = 17 + 25(2 + 5t) = 17 + 50 + 125t = 67 + 125t $.

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