# CBSE Class 10 Answered

**From each corner of a square sheet of side 8cm, a square of side y cm is cut. The remaining sheet is folded into a cuboid. The minimum possible volume of the cuboid formed is M cubic cm. If y is an integer, then find M.**

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 y^{2} + 4y^{3}** **) ................................ (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+12y^{2} = 0

Hence , 3y^{2} - 16y + 16 = 0

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

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

d^{2}V/dy^{2} = 24y-64

( d^{2}V/dy^{2}) at y = 4/3 cm is -32 . hence at y = 4/3 cm volume is maximum

( d^{2}V/dy^{2}) 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}