1. ## Extreme Value Problems

A box with an open top is to be constructed from a square piece of cardboard, 6 meters wide, by cutting out a square from each of the four corners, and bending up the sides. Find the largest volume that such a box can have.

Let x represent the dimensions

$\displaystyle V = (6 - 2x)^2x$
$\displaystyle V = 36x - 24x^2 + 4x^{3}$

$\displaystyle V' = 36 - 48x + 12x^2$
$\displaystyle 0 = 36 - 48x + 12x^2$
$\displaystyle 0 = 12(x - 1)(x - 3)$

$\displaystyle x = 1 and x = 3$

Restrictions: $\displaystyle 0 \leq x \leq 3$

$\displaystyle V = Lwh$
$\displaystyle V = (3)(1)(6)$
$\displaystyle V = 18m^{3}$

---------

Text Answer is $\displaystyle 16m^{3}$

2. Originally Posted by Macleef
A box with an open top is to be constructed from a square piece of cardboard, 6 meters wide, by cutting out a square from each of the four corners, and bending up the sides. Find the largest volume that such a box can have.

Let x represent the dimensions

$\displaystyle V = (6 - 2x)^2x$
$\displaystyle V = 36x - 24x^2 + 4x^{3}$

$\displaystyle V' = 36 - 48x + 12x^2$
$\displaystyle 0 = 36 - 48x + 12x^2$
$\displaystyle 0 = 12(x - 1)(x - 3)$

$\displaystyle x = 1 and x = 3$

Restrictions: $\displaystyle 0 \leq x \leq 3$

$\displaystyle V = Lwh$
$\displaystyle V = (3)(1)(6)$
$\displaystyle V = 18m^{3}$

---------

Text Answer is $\displaystyle 16m^{3}$

$\displaystyle V(x) = (6 - 2x)^2x$
$\displaystyle V(1) = (6 - 2(1))^2(1)=(4)^2=16$