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Question: A glaciers ice thickness decreases by $ x $ meters annually, and its area shrinks by $ x + 2 $ square kilometers. If the total loss over 3 years is 45 square kilometers, find $ x $.
Question: A glaciologist measures ice layer thickness $ t $ and snow accumulation $ s $, satisfying $ 2t + 5s = 22 $ and $ 3t - s = 4 $. Find $ t + s $.
The equation of the perpendicular line passing through $ (5, 7) $ is:
Set the two expressions for $ y $ equal:
Multiply through by 12 to eliminate denominators:
Question: A regular tetrahedron is embedded in 3D space with three of its vertices at $ A(1, 1, 1) $, $ B(1, -1, -1) $, and $ C(-1, 1, -1) $. The fourth vertex $ D $ has integer coordinates. Find the coordinates of $ D $.
Let the fourth vertex be $ D(x, y, z) $. Since the tetrahedron is regular, all edges must have equal length.
First, compute the squared distance between $ A $ and $ B $:
Only integer solution is $ x = -1 $. Then $ y = x = -1 $, $ z = -x = 1 $
Wait—this yields no real solution. But check computation: