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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.
Asked by vandana | 17 Feb, 2024, 12:46: PM

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

Answered by Thiyagarajan K | 17 Feb, 2024, 03:20: PM
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