\boxed{10\pi x^2}

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• The lateral surface area of a cone is given by:
• where $ r = 2x $ and $ l = 5x $. Substituting:
• A = \pi (2x)(5x) = 10\pi x^2
• 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}

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