Question

A square piece of cardboard is to be used to form a box without a top by cutting off squares, 3 cm on a side from each corner and then folding up the sides. The volume of the box must be 3000cm^3

Answer 1

Step-by-step explanation:

Let the length of the each side of the square which is cut from each corner of the tin sheet be x cm.

By folding up the flaps, a cuboidal box is formed whose length, breadth and height are 18−2x,18−2x and x respectively.

Then, its volume V is given by

V=(18−2x)(18−2x)x

⇒

dx

dV

=324−144x+12x

2

and

dx

2

d

2

V

=−144+24x

The critical numbers of V are given by

dx

dV

=0

⇒324−144x+12x

2

=0

⇒x

2

−12x+27=0

⇒x=3,9.

But, x=9 is not possible. Therefore, x=3.

Clearly, (

dx

2

d

2

V

)

x=3

=−144+72=−72<0.

So, V is maximum when x=3 i.e., the length of each side of the square to

be cut is 3cm.