Question

A sheet of cardboard 12 inches square is used to make an open box by cutting out squares of equal size from the four corners and bending up the sides. What size should the squares be to obtain a box with the largest possible volume

Answer 1

Answer:

$x=6+3\sqrt{2}$ is the size of the square

Step-by-step explanation:

Let the side of the square cut from the cardboard be “x”

Size of the square base = $12 -2x$, height of the box = x

Volume of square box = Area * height = $(12-2x)^2 * x$

Differentiating the above equation and equating it to zero, we get –  

d/dx $((12-2x)^2 * x)$

d/dx${(144 + 4x^2 -48x)*x}$= d/dx $(144x + 4x^3 -48x^2)$

d/dx $(144x + 4x^3 -48x^2) = 0$

$144+8x^2-96X= 0\\18 +X^2-12X = 0$

On solving above equation, we get –  

$x=6+3\sqrt{2} \\ x=6-3\sqrt{2}$