Question

Q3. 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 fold the paper at the cuts to from an open box
with maximum volume. What should be the size of squares cut so that the volume of
the open box is maximum.​

Answer 1

Answer:

Size of square = (5/3) * (5/3)  

Step-by-step explanation:

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 fold the paper at the cuts to from an open box  with maximum volume. What should be the size of squares cut so that the volume of  the open box is maximum.​

Let say Size of square cut = x * x cm

Then height of Open box = x cm

Length & Breadth of open box = 10 - 2x  cm

Volume of box = (10 - 2x)(10 -2x)x  cm³

= ( 100 + 4x² - 40x)x

= 100x + 4x³ - 40x²

dV/dx  =  100 + 12x² - 80x

dV/dx  = 0

12x² - 80x + 100 = 0

=> 3x² - 20x + 25 = 0

=> 3x² - 15x - 5x + 25 = 0

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

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

=> x = 5/3  x = 5

d²V/dx² = 24x - 80

for x = 5  d²V/dx² = 40 ( +ve) => x = 5 will give minimum volume

for x = 5/3 d²V/dx² = -40 ( -ve) => x = 5/3 will give maximum volume

Size of square cut = (5/3) * (5/3)

Answer 2

Answer: 64 cm^3

Step-by-step explanation:

Length of the box = 10 – (1+1)

= 10 -2

= 8 cm

Breadth of the box = 10 – (1+1)

= 10 – 2

= 8 cm

Height of the box = 1 cm

Volume of the open box = l*b*h

= 8*8*1

= 64 cm^3