Share

# A Square Piece of Tin of Side 18 Cm is to Be Made into a Box Without Top by Cutting a Square from Each Corner and Folding up the Flaps to Form a Box. What Should Be the Side of the Square - CBSE (Arts) Class 12 - Mathematics

#### Question

A square piece of tin of side 18 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Find this maximum volume.

#### Solution

Let the side of the square to be cut off be x cm.

Then, the length and the breadth of the box will be (18 − 2x) cm each and height of the box will be x cm.

Volume of the box, V(x) = x(18 − 2x)2

$V'\left( x \right) = \left( 18 - 2x \right)^2 - 4x\left( 18 - 2x \right)$

$= \left( 18 - 2x \right)\left( 18 - 2x - 4x \right)$

$= \left( 18 - 2x \right)\left( 18 - 6x \right)$

$= 12\left( 9 - x \right)\left( 3 - x \right)$

$V''\left( x \right) = 12\left( - \left( 9 - x \right) - \left( 3 - x \right) \right)$

$= - 12\left( 9 - x + 3 - x \right)$

$= - 24\left( 6 - x \right)$

$\text { For maximum and minimum values of V, we must have }$

$V'\left( x \right) = 0$

$\Rightarrow$ x = 9 or x = 3

If x = 9, then length and breadth will become 0.

∴ x ≠ 9

$\Rightarrow$ x = 3

Now,

$V''\left( 3 \right) = - 24\left( 6 - 3 \right) = - 72 < 0$

∴  x = 3 is the point of maxima.

$V\left( x \right) = 3 \left( 18 - 6 \right)^2 = 3 \times 144 = 432 {cm}^3$

Hence, if we remove a square of side 3 cm from each corner of the square tin and make a box from the remaining sheet, then the volume of the box so obtained would be the largest, i.e. 432 cm3

Is there an error in this question or solution?

#### Video TutorialsVIEW ALL [1]

Solution A Square Piece of Tin of Side 18 Cm is to Be Made into a Box Without Top by Cutting a Square from Each Corner and Folding up the Flaps to Form a Box. What Should Be the Side of the Square Concept: Graph of Maxima and Minima.
S