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PROPRIETARY MATERIAL. © 2013 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, 
reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited 
distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, 
you are using it without permission. 
630 
PROBLEM 13.82* (Continued) 
 
 Thus, 2 2 1/2
2 2 1/2 00
2 2 1/2 2 1/2
( 2 )
( 2 )
( 2 ) (2 ) ( 3 2)
a a
BD
BD
OABD O A A B B D
y
U dy y a
y a
U a a a a
U U U U− − −
= = +
+
= + − = −
= + +
 
 ( 2 1) ( 3 2)a a a= + − + − 3OABDU a=  
 2 2 2 1/2
( , , ) (0,0,0)
( ) 0
ODV V a a a V
a a a
Δ = −
= − + + − 3ODV aΔ = −  
 Thus, OABD ODU V= −Δ

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