CBSE Class 10 Answered

side length of given square sheet = 8 cm

if square of y cm is cut in each corner and cuboid is formed , then volume V of cuboid as a function of y is

V(y) = ( 8-2y)² y = ( 64 y - 32 y² + 4y³ )  ................................ (1)

To get the value of y that corresponds to minimum or maximum volume, we differentiate above function and equate it to zero.

dV/dy = 64-64y+12y² = 0

Hence , 3y² - 16y + 16 = 0

(y-4)(3y-4) = 0

hence maximum and minimum or occuring at y = 4 cm  or y = 4/3 cm

d²V/dy² = 24y-64

(  d²V/dy²) at y = 4/3 cm is  -32 . hence  at y = 4/3 cm  volume is maximum

(  d²V/dy²) at y = 4 cm  is  32 . hence  at y = 4 volume is minimum

If we substitute y = 4 cm in eqn.(1) , we get V = 0

since y is integer , then y = 3 cm  gives the minimum volume

Let us substitute y = 3 cm in eqn.(1) to get minimum volume V as

V = ( 8- 6)² × 2  (cm)³ = 8 (cm)³