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Home » 2Question: In spherical coordinates $( - Richter Guitar

2Question: In spherical coordinates $( - Richter Guitar

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 Solution: The rectangular array is inscribed in a circle of radius 10 km, so the diagonal of the rectangle equals the diameter of the circle:

📌 Let the sides be 8 km and 15 km. Then, the diagonal $ d $ satisfies:

📌 Unless the array is not required to have orthogonal sides? But rectangles do.

📌 Question: A spherical energy storage unit has radius $ r $ meters, while a hemispherical backup unit has radius $ 3r $ meters. What is the ratio of the volume of the spherical unit to the volume of the hemispherical unit?

📌 Solution: Volume of a sphere of radius $ r $ is:

📌 Volume of a hemisphere of radius $ 3r $ is half the volume of a full sphere of that radius:

📌 Question: A materials scientist models a crystal lattice with a regular tetrahedron. Three vertices are at $(1, 2, 3)$, $(4, 5, 6)$, and $(7, 8, 9)$. Find the integer coordinates of the fourth vertex.

📌 Solution: The centroid of a regular tetrahedron is the average of its vertices. Let the fourth vertex be $(x, y, z)$. The centroid is:

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