Question 224025
The bottom of a box is supposed to be a rectangle with the perimeter of 36 cm. The box must be 4 cm high. what dimensions give the maximum volume?


Step 1.  The maximum volume is when the rectangle is a square and let s be the side of the square.


Step 2.  The perimeter P of a rectangle (or square) means adding up all four sides.  So P=s+s+s+s=4s since all four sides of the square are equal.


Step 3.  But P=36=4s.  Solving for s=36/4=9 cm.


Step 4.  Volume V=A*h of the box where A is the area of the square and h is the height given as h=4.


Step 5.  Area {{{A=s^2}}} for a square.  So {{{A=9^2=81}}} square centimeters.


Step 6.  Then V=A*h=81*4=324 cubic centimeters.


Step 7.  ANSWER:  The maximum volume of the box is 324 cubic centimeters.


I hope the above steps and explanation were helpful. 


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