Pre-Calc Homework Solutions 156

# Pre-Calc Homework Solutions 156 - 156 Section 4.4 x 4 9 Let...

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7. Let x be the side length of the cut-out square (0 , x , 4). Then the base measures 8 2 2 x in. by 15 2 2 x in., and the volume is V ( x ) 5 x (8 2 2 x )(15 2 2 x ) 5 4 x 3 2 46 x 2 1 120 x . Then V 9 ( x ) 5 12 x 2 2 92 x 1 120 5 4(3 x 2 5)( x 2 6). Then the critical point (in 0 , x , 4) occurs at x 5 } 5 3 } . Since V 9 ( x ) . 0 for 0 , x , } 5 3 } and V 9 ( x ) , 0 for } 5 3 } , x , 4, the critical point corresponds to the maximum volume. The maximum volume is V 1 } 5 3 } 2 5 } 24 2 5 7 0 } < 90.74 in 3 , and the dimensions are } 5 3 } in. by } 1 3 4 } in. by } 3 3 5 } in. Graphical support: [0, 4] by [ 2 25, 100] 8. Note that the values a and b must satisfy a 2 1 b 2 5 20 2 and so b 5 ˇ 4 w 0 w 0 w 2 w a w 2 w . Then the area is given by A 5 } 1 2 } ab 5 } 1 2 } a ˇ 4 w 0 w 0 w 2 w a w 2 w for 0 , a , 20, and } d d A a } 5 } 1 2 } a 1 } 2 ˇ 4 w 0 w 1 0 w 2 w a w 2 w } 2 ( 2 2 a ) 1 } 1 2 } ˇ 4 w 0 w 0 w 2 w a w 2 w 55 } ˇ 2 4 w 0 0 w 0 0 w 2 2 w a a w 2 2 w } . The critical point occurs when a 2 5 200. Since } d d A a } . 0 for 0 , a , ˇ 2 w 0 w 0 w and } d d A a } , 0 for ˇ 2 w 0 w 0 w , a , 20, this critical point corresponds to the maximum area. Furthermore, if a 5 ˇ 2 w 0 w 0 w then b 5 ˇ 4 w 0 w 0
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## This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Spring '08 term at University of Florida.

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