Question

A square tin sheet of side $12$ inches is converted into a box with an open top in the following steps the sheet is placed horizontally. Then, equal sized squares, each side $x$ inches, are cut from the four corners of the sheet. Finally, the four resulting sides are bent vertically upwards in the shape of a box. If $x$ is an integer, then what value of x maximizes the volume of the box?

Answer 1

Hint : In this question we have to find the volume of the cuboid and differentiate it to get the value of $x$ maximizes the volume of the box. Also apply the method of factorization.

Complete step-by-step answer :
After conversation sheet in the form of a box it becomes a cuboid so

Length of the box will be $= 12 - x - x$
$= 12 - 2x$ Inches
Breadth of the box will be $= 12 - x - x$
$= 12 - 2x$ Inches
Height of the box will be $= x$ Inches
We know that volume of cuboid $= length \times breadth \times height$
$= \left( {12 - 2x} \right) \times \left( {12 - 2x} \right) \times x$
$= \mathop {\left( {12 - 2x} \right)}\nolimits^2 x$
$= \left( {144 + 4\mathop x\nolimits^2 - 2 \times 12 \times 2x} \right)x$
$= 144x + 4{x^3} - 48{x^2}$
Here volume of the cuboid $V = 144x + 4{x^3} - 48{x^2}$
Now differentiation of volume of the cuboid with the respect of $x$ we get.
$\dfrac{{dv}}{{dx}} = \dfrac{d}{{dx}}\left( {144x + 4\mathop x\nolimits^3 - 48\mathop x\nolimits^2 } \right)$
$= \dfrac{d}{{dx}}144x + \dfrac{d}{{dx}}4\mathop x\nolimits^3 - \dfrac{d}{{dx}}48\mathop x\nolimits^2$
$= 144\dfrac{d}{{dx}}x + 4\dfrac{d}{{dx}}\mathop x\nolimits^3 - 48\dfrac{d}{{dx}}\mathop x\nolimits^2$
$= 144 + 12\mathop x\nolimits^2 - 96x$
$= \mathop x\nolimits^2 - 8x + 12$
After factorization of this we get.
$x = 6,2$
Putting $x = 6$ in the volume of cuboid we get,
$= 144 \times 6 + 4 \times \mathop {\left( 6 \right)}\nolimits^3 - 48 \times \mathop {\left( 6 \right)}\nolimits^2$
$= 0$
Putting $x = 2$ in the volume of cuboid we get
$= 144 \times 2 + 4\mathop {\left( 2 \right)}\nolimits^3 - 48\mathop {\left( 2 \right)}\nolimits^2$
$= 128$ Inches
Hence the value of $x$ in this question is $2$ and volume of the tin sheet is $128$ inches
So, the correct answer is “ $x$ in this question is $2$ and volume of the tin sheet is $128$ inches”.

Note : In this question the bent shape of the box looks like a cuboid. A cuboid is a three-dimensional shape which has six faces, which form a convex polyhedron. Be careful while understanding the word statements and letters, read it twice, do simplification and solve using the mathematical operations.