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word problem will rate lifesaver please help..difficult problem..need help

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Date Posted: 7/24/2008 4:06:02 PM  Status: Live
word problem will rate lifesaver please help..difficult problem..need help
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Question Details:
 a manufacturer constructs open boxes from sheets of cardboard that are 6 inches square by cutting small squartes from the corners and flding up ther sides. the research and development department asks you to determine the size of the square that produces a box of the greatest volume.
a= let x be the length of a side of the square to be cut, and let V be the volume of the resulting box. show that V= x(6-2x)2.
b= are there any restrictions on the value of x?..explain
c= make a graph of V versus x over an appropiate interval, and use the graph to estimete the value of x that results in the largestvolume.
d= estimate the largest volume

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Date Posted: 7/24/2008 4:45:36 PM  Status: Live
Asker's Rating: Lifesaver   
Response:
a) Call the length of a side of the square corner that's cut out x.
For the box, when a square is removed at each corner, the length of the remaining side flap will be (6 - 2x).  V = L * W * H = (6 - 2x)(6-2x)(x).

b) There are restrictions on the value of x.  Examine the equation for the volume of the box above.  If x <= 0 or if x >= 3, then there will be negative or zero dimensions and a negative or zero volume.  So,

c) 


d)  dV/dx = 12x^2 - 48x + 36

Set this equal to zero to calculate the maximum possible volume.

12x^2 - 48x + 36 = 0 = (x-3)(x-1)

By the restrictions above, we know that x cannot be 3.  This leaves the maximum volume when x = 1.  V(1) = 16.  This is confirmed by the graph.
kwk's Comment:
thanks alot...good work




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