Question

An open box is to be made from a square pieces of material, 24cm on a side by cutting equal square from the corner and up the sides as shown express the volume V of the box as a function of x​

Answer 1

Hi fathima. Here is your answer.

Answer:

An open box is to be made from a square piece of material, 24 cm on a side, by cutting equal squares from the corners and turning up the sides.

Size of square piece = 24 * 24  cm²

Let say Size of square cut from corner = x * x   cm²

Then sides of Open box would be be

24 - 2x  , 24 - 2x  & 2x

Volume of the open box = (24-2x)(24-2x)2x

= 2x(24 - 2x)²

= 2x * 2²(12 - x)²

= 8x * (x² + 144 - 24x)

= 8x³ - 192x + 1152x

V = 8x³ - 192x² + 1152x

dV/dx = 24x² - 384x + 1152

dV/dx  = 0

24x² - 384x + 1152  = 0

x² - 16x + 48 = 0

x² - 12x - 4x + 48 = 0

x(x-12) -4(x-12) = 0

(x - 4)(x-12) = 0

x = 12 is not possible as then no box will be left

x = 4 will give max volume

Hope this helps!

Answer 2

Answer :

The volume of the box expressed as a function of x is:

V = 4x³ - 96x² + 576x

Step-by-step explanation :

Given that :

Initial length of the square piece = 24.

Length of the box = 24-2x

Breadth of the box = 24 - 2x

Height of the box = x

To Find :

The volume V of the box as a function of x​

Solution :

After cutting "x" cm from each corner.

The remaining dimension of the box can be :

Length of the box = 24-2x

Breadth of the box = 24 - 2x

Height of the box = x

We know that :

$\bigstar\; \boxed{\sf Volume = L\times B \times H}}$

Put the given values :

$\sf \implies (24-2x)(24-2x)x\\\\\implies x(24-2x)^2\\\\\ \implies x( 576 + 4x^2 - 96x)\\\\\implies V = 4x^3 - 96x^2 + 576x$

Hence,

The volume of the box expressed as a function of x is:

V = 4x³ - 96x² + 576x.

$\bigstar\; \textbf{\underline{Refer the attachment for the figure.}}$