Optimization

Timothy Ornelas's picture
Submitted by Timothy Ornelas on March 16, 2015 - 10:19pm

An open box is to be made out of a 10-inch by 18-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume.

I need the dimensions of the bottom of the box and the height of the box.

Please help

  • Log in to post comments
  • 7006 reads
Jhun Vert's picture

Submitted by Jhun Vert on March 18, 2015 - 3:43pm.

Length = 18 - 2x
Width = 10 - 2x
Height = x
 

win_20150319_155824.jpg

 

Volume = Length × Width × Height
$V = (18 - 2x)(10 - 2x)(x)$

$V = 180x - 56x^2 - 4x^3$

$\dfrac{dV}{dx} = 180 - 112x - 12x^2 = 0$

$x = 1.3978 ~ \text{ and } ~ -10.7311$
 

Use x = 1.4 inches

Dimensions of the largest box:
Length = 18 - 2(1.4) = 15.2 inches
Width = 10 - 2(1.4) = 7.2 inches
Height = 1.4 inches

Note: I did not re-check my solution.