Mathamatics

(Full name (Professor (Course (18 December 2006 (Titlea ) The U .S . Postal Service pull up stakes accept a cuff for domestic shipment totally if the sum of its aloofness and girth (distance around ) does not take place 108 in (The last time I called the post mail service this was still true Suppose that the box has a unanimous end ( See in the Picture . What dimensions will ante up a box of maximum volumeLet L continuance of boxx cheek of square end108 length girth108 L 4xL 108 - 4x p Volume f (x (length (width (height (L (x (x (108 - 4x (x2 108x2 - 4x3The run short is therefore : f (x 108x2 - 4x3Domain of x : [0 , 108]We find the detailed numbers (first derivative ) to know the relation extremaf (x 108x2 - 4x3f (x (2 (108 )x - (4 (3 )x2We set the order to zero point to find the vital numbers0 (2 (108 )x - (4 (3 )x20 216x - 12x20 x (216 - 12xCritical numbers : x 0 or 18Although x 0 is in the public , the side of a box drop t be zero so x could only be 18x 18To check if at x 18 , the y value is a congress maximum , we find the second derivativef (x 108x2 - 4x3f (x 216x - 12x2f (x 216 - (2 (12 )x 216 - 24xWe substitute 18 to the comparability 216 - (24 (18 -216 ( The value is negative , which means that at the point where x 18 , the function value is at a relative maximum .

It is also the absolute maximum value because the determine for the side of the box x 0 and x 108 can t be trueNow that we have x , we now find the length of the boxL 108 - 4x 108 - (4 (18 36Thus , the dimensions of a box of maximum value are : length x width x height 36 x 18 x 18b ) Suppose that instead of having a box with square ends you have a box with square sides (See the control ) What dimensions will give the box of largest volumeLet L length of boxX former(a) side108 length girth108 L (2L 2x108 3L 2xL (108 - 2x / 3Volume f (x (length (width (height (L (x (L L2xDomain of x : [0 , 108]To find the critical numbers , we find the first derivative of the function and gibe it with 0f ` (x (x (20 (27 - 2 /3x )2 - 4 /3x (27 - 2 /3x0 272 - 36x 4 /9x2 - 36x - 8 /9x20 272 - 72x 4 /3x2x 40 .5 , 13 .5We find the second derivative of the function to find out which of the two is at a relative maximumf ` (x 362 - 96x 4 /3x2f (x (2 (4 /3 )x - 96 8 /3x - 96Case 1 : x 40 .5f (x (8 /3 (40 .5 ) -...If you want to get a complete essay, order it on our website: Orderessay

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