# Thread: Need help with optimization(differentiation)

1. ## Need help with optimization(differentiation)

A rectangular piece of cardboard with dimension 12 cm by 24 cm is to be made into an open box (i.e., no lid) by cutting out squares from the corners and then turning up the sides. Find the size of the squares that should be cut out if the volume of the box is to be a maximum.

2. Originally Posted by ninja
A rectangular piece of cardboard with dimension 12 cm by 24 cm is to be made into an open box (i.e., no lid) by cutting out squares from the corners and then turning up the sides. Find the size of the squares that should be cut out if the volume of the box is to be a maximum.
HI

The volume would be ,

$\displaystyle V=(12-2x)(24-2x)(x)$

$\displaystyle =288x-72x^2+4x^3$

$\displaystyle \frac{dV}{dx}=288-144x+12x^2$

$\displaystyle 288-144x+12x^2=0$

Then here use the quadratic formula to solve for x .

$\displaystyle \frac{d^2V}{dx^2}=24x-144$

Substitute both values of x u got into this equation , whichever gives you $\displaystyle <0$ (since it is maximum) will be the answer .

Then the size would be this $\displaystyle x^2$