Question

A square piece of tin of side $18cm$ 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 then box is maximum? Also find this maximum volume?

Answer 1

Let the side of the square pice cut from each corner of the given square plate of side $18cm$ be $xcm$.

Then the dimension of the open box are $(18-2x)cm,(18-2x)cm$ and $xcm$

$\therefore V=(18-2x{)}^2\times x$

$\Rightarrow V=4x^3-72x^2+324x$

$\Rightarrow \frac{dV}{dx}=12(x^2-12x+27)$ and $\frac{d^{2}V}{d{x}^2}=12(2x-12)$.

Now, $\frac{dV}{dx}=0\iff x^2-12x+27=0\iff (x-3)(x-9)=0\iff x=3[\because x\ne 9]$ and ${\left[\frac{d^{2}V}{d{x}^2}\right]}_{(x=3)}=12(2\times 3-12)=-72<0$.

$\therefore V$ is maxima at $x=3cm$, and maximum value $=[4\times 3^3-72\times 324\times 3]c{m}^3=432c{m}^3$.