Pre-Calc Homework Solutions 158

Pre-Calc Homework Solutions 158 - 158 Section 4.4 15 2 2x...

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18. (a) The base measures 10 2 2 x in. by } 15 2 2 2 x } in., so the volume formula is V ( x ) 55 2 x 3 2 25 x 2 1 75 x . (b) We require x . 0, 2 x , 10, and 2 x , 15. Combining these requirements, the domain is the interval (0, 5). [0, 5] by [ 2 20, 80] (c) [0, 5] by [ 2 20, 80] The maximum volume is approximately 66.02 in 3 when x < 1.96 in. (d) V 9 ( x ) 5 6 x 2 2 50 x 1 75 The critical point occurs when V 9 ( x ) 5 0, at x } 50 6 1 ˇ 2 7 w 0 w 0 w } 5 } 25 6 6 5 ˇ 7 w } , that is, x < 1.96 or x < 6.37. We discard the larger value because it is not in the domain. Since V 0 ( x ) 5 12 x 2 50, which is negative when x < 1.96, the critical point corresponds to the maximum volume. The maximum volume occurs when x 5 } 25 2 6 5 ˇ 7 w } < 1.96, which confirms the result in (c). 19. (a) The “sides” of the suitcase will measure 24 2 2 x in. by 18 2 2 x in. and will be 2 x in. apart, so the volume formula is V ( x ) 5 2 x (24 2 2 x )(18 2 2 x ) 5 8 x 3 2 168 x 2 1 864 x . (b) We require x . 0, 2 x , 18, and 2 x , 24. Combining these requirements, the domain is the interval (0, 9).
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