Question

70 points if u answer the question!!pls I need help!!!

Take a square sheet of paper of side 10cm. Four small squares are to be cut from the corners of the square sheet and then folded at the cuts to form an open box .What should be the size of the squares cut so that the volume of the open box is maximum.

Answer 1

let side length of each square = x cm

then length of box =( 10-2x ) cm
width of box = ( 10 -2x) cm
height of box = x cm

so, volume of box ( V) = l × b × h
= (10-2x )( 10-2x )x cm³

V(x ) = (100 -40x + 4x²)x
= 100x -40x² + 4x³

differentiate wrt x
dv/dx = 100 -80x +12x² = 0
3x² - 20x +25 = 0

3x² -15x -5x +25 =0

(3x -5)(x -5) = 0

x = 5/3 and 5

d²V/dx² < 0 at x = 5/3

so, at x = 5/3 volume will be maximum

so, x = 5/3 cm

Answer 2

Well!! Just draw in a paper and see!!!!!!! 
There would be four sides extending. 
Base will be a square of side (10-2x). 
Hence, base area=(10-2x)^2 
Remaining four cornered segments are rectangles of (10-2x)x 
Height= x 
Volume of the box = Base area x Height 
v= x(10-2x)^2 
= x(2^2)(5-x)^2 
= 4x(25-10x+x^2) 
=4(25x-10x^2+x^3) 
For max vol, dv/dx =0 
(i.e) 25-20x+3x^2=0 
3x^2-15x-5x+25=0 
3x(x-5)-5(x-5) =0 
(x-5)(3x-5)=0 
Hence, x=5,5/3 

Now, for max, d^2v/dx^2 <0 
6x-20<0 
x=5 implies value=5>0 
x=5/3 implies value = -10<0 


Hence, the side of square cut off=5/3 =1.66 cm