You have a open box that is made from a 16 in. x30 in. piece of cardboard. When you cut out the squares of equal size from the 4 corners and bending it. What size should the squares be to get this box to work with the largest volume?

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Answer 1 · 2018-03-04T12:30:50

# 3 1/3# inches to be cut from #4# corners and bend to get
box for maximum volume of #725.93# cubic inches.

Card board size is #L=30 and W=16# inches

Let #x# in square is cut from #4# corners and bended into

a box whos size is now #L=30-2x , W=16-2x and h=x#

inches. Volume of the box is #V=(30-2x)(16-2x)x# cubic

inches. #V=(4x^2-92x+480)x = 4x^3-92x^2+480x#.

For maximum value #(dV)/dx=0#

#(dV)/dx=12x^2-184x+480=12(x^2-46/3x+40)#

#12(x^2-12x-10/3x+40)= 12(x(x-12)-10/3(x-12))#

or #12(x-12)(x-10/3)=0 :.# Critical points are

#x=12 ,x=10/3; x !=12# , as #24# inches cannot be removed from

# 16 # inches width. So #x= 10/3 or 3 1/3# inches to be cut.

Slope test may be examined at#(x=3 and x=4)# to show

volume is maximum. #(dV)/dx=12(x-12)(x-10/3)#

#(dV)/dx(3)= (+) and (dV)/dx(4)= (-)#. Slope at critical point

is from positive to negative , so the volume is maximum.

The maximum volume is #V=(30-20/3)(16-20/3)10/3#or

#V=(30-20/3)(16-20/3)10/3 ~~725.93# cubic inches. [Ans]