a1 solu - MATH 251-3, Fall 2007 Simon Fraser University...

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MATH 251-3, Fall 2007 Simon Fraser University Assignment 1: Solutions Additional Question Due: 4:30pm, Monday 17 September 2007 1. Describe clearly in words the geometry of the surface or region in R 3 described by the graph of the following equation or inequality: (a) x 2 + y 2 + z 2 + 5 y + 31 = 10 z . Solution: We first rearrange terms and complete the square to get the equation in standard form: x 2 + y 2 + 5 y + z 2 - 10 z + 31 = 0 = x 2 + y 2 + 5 y + 25 4 + z 2 - 10 z + 25 = - 31 + 25 4 + 25 = x 2 + ± y + 5 2 ² 2 + ( z - 5) 2 = 1 4 = ± 1 2 ² 2 . From this formula, we can read off that the equation represents the surface of a sphere, centre at ( x 0 , y 0 , z 0 ) = (0 , - 5 2 , 5) and radius 1 2 . (b) x 2 + z 2 9 . Solution: Note that this inequality is independent of y ; that is, the geometrical object de-
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This note was uploaded on 10/09/2009 for the course MATH macm 101 taught by Professor Jcliu during the Spring '09 term at Simon Fraser.

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