Question

A rectangular box is to be made from a sheet of $24$ inch length and $9$ inch width cutting out identical squares of side length x from the four corners and turning up the sides.

What is the value of x for which the volume is maximum?

• A. $1$ inch
• B. $1.5$ inch
• C. $2$ inch
• D. $2.5$ inch

Answer 1

The correct option is C $2$ inch

Length of the Cuboid = $24-2x$

Breadth of Cuboid =$9-2x$

Height of Cuboid =$x$

Volume of Cuboid $V=(24-2x)(9-2x)x$

$V=216x+4x^3-66x^2$

To know the volume maximum $\frac{dV}{dx}=0$

$\Rightarrow \frac{dV}{dx}=216+12x^2-132x=0$

$\Rightarrow \frac{dV}{dx}=18+x^2-11x=0$

$(x-9)(x-2)=0$

$x=9or2$

By substituting 9 and 2 we get maximum value of volume for $x=2$inch