Question

.    An open box is to be made from a square piece of material 18cm on a side by cutting equal squares from the corners and turning up the sides. 

    (a) Write down the volume V of the box as a function of its height x.

    (b) What is the domain of the function?

    (c)  Determine the volume of a box with a height of 4cm.. ​

Answer 1

Step-by-step explanation:

Let each side of the square cut off from each corner be x cm

Then the base of the box will be of side 18−2x cm and the height of the box will be x cm

Then volume of box V=(18−2x)(18−2x)x

V=(18−2x)

2

x

V=4x

3

+324x−72x

2

...(i)

Differentiating w.r t to x, we get

dx

dV

=12x

2

+324−144x

dx

dV

=12(x

2

−12x+27) ....(ii)

For maximum volume

dx

dV

=0

⇒12(x

2

−12x+27)=0

⇒ x

2

−9x−3x+27=0

⇒ (x−9)(x−3)=0

⇒ x=9,3

Again differentiating, we get

dx

2

d

2

V

=2x−12 ...(iii)

At x=9,

dx

2

d

2

V

=+ve

∴ V is minimum at x=9 at x=3

dx

2

d

2

V

=−ve

∴ V is maximum at x=3

∴ Maximum volume V=(18−6)(18−6)×3

=12×12×3=432cm

3