SOLUTION: Point P is the center of one face of a cube and Point Q is the center of the opposite face. If the length of the shortest possible path from P to Q along the outer surface of the c

Algebra ->  Volume -> SOLUTION: Point P is the center of one face of a cube and Point Q is the center of the opposite face. If the length of the shortest possible path from P to Q along the outer surface of the c      Log On


   

Question 125148: Point P is the center of one face of a cube and Point Q is the center of the opposite face. If the length of the shortest possible path from P to Q along the outer surface of the cube is 2(the square root of 2)cm, what is the volume of the cube in cubic centimeters?
Answer by solver91311(24713) About Me  (Show Source):
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The shortest path around the outside of the cube is perpendicular to the nearest edge of the cube, straight across the next face through its center, and then perpendicular to the edge of the cube to the next point.

The first part of the path is exactly half of the length of an edge, the second part is exactly the length of an edge, and the third part is again half the length of an edge. So the entire path is 2 times the length of an edge. But we are given that the length of the path is 2%28sqrt%282%29%29, therefore the length of one edge is sqrt%282%29.

The volume is the length of an edge cubed, so V=%28sqrt%282%29%29%5E3=red%282%2Asqrt%282%29%29