\boxed{x\sqrt{3}}

February 22, 2026

$\boxed{x\sqrt{3}}$

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Question: A quantum algorithm designer is modeling a 3D lattice where each node is a cube of side $ x $. A diagonal connects two opposite vertices of the cube. What is the length of this space diagonal in terms of $ x $?
The space diagonal $ d $ of a cube with side length $ x $ is given by the 3D Pythagorean theorem:
d = \sqrt{x^2 + x^2 + x^2} = \sqrt{3x^2} = x\sqrt{3}
Question: A marine safety consultant studies a triangular coral reef formation with sides of 10 m, 10 m, and 12 m. What is the radius of the inscribed circle in meters?
Let $ a = 10 $, $ b = 10 $, and $ c = 12 $. The semi-perimeter is:
s = \frac{10 + 10 + 12}{2} = 16

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