Question

an open box with a square base is to be made out of a given iron sheet of Area 27 square metres show that the maximum volume of the box is 13.5cu.cm

​

Answer 1

Let 'x'be the side of the Square base and 'y' the height of the box.

Area of the square base = x²

and area of four walls = 4xy

by the question,

x²+4xy=27.....(1)

Now V ,

volume of the box= (area the base)×(height)

volume of the box= x²y=x²(27-x²)/4x

= 1/4(27x-x³).....(2)

dV/dx= 1/4 (27-3x²).....(3)

and

dV/dx= -3/2x.......(4)

Now,

dV/dx= 0 gives

1/4(24-3x²)=0=>x²=27/3=9

=>x=±3

=>x=3 [x can't be negative]

V can only max. when x= 3m

when x=3 from (4),

d²V/dx²=-3×3/3= -9/2

which is negative

Thus x= 3 gives max. value of V

Hence,

max. V= 1/4[27× 3-27] [ putting x=3 in 2]

=27/4(3-1)

= 27/2

= 13.5 cm³.