Question

You are constructing a cardboard box with the dimensions 2 m by 4 m. You then cut equal-size squares ( x by x squares) from each corner so that you may fold the edges to get a box. How long should x be to get a box with the maximum volume?Round your answer to three decimal places

Answer 1

Answer:

x = 0.423 m

Step-by-step explanation:

Volume of the box = length × width × height

The height is given by the length of a side of the square.

height = x

The width and height are gotten by subtracting twice the side of the square form each dimension of the cardboard (since two squares are cut along each dimension).

length = 4 - 2x

width = 2 - 2x

Volume, $V = x(4-2x)(2-2x) = 8x - 12x^2 + 4x^3$

To find the maximum, we differentiate V with respect to x and equate the derivative to 0.

$\dfrac{dV}{dx} = 8-24x+12x^2 = 0$

Simplify by dividing by 4.

$2-6x+3x^2 = 0$

Using the quadratic formula,

x = 0.423 m or x = 1.577 m

To determine which values gives maximum, we differentiate the first derivative which gives

$\dfrac{d^2V}{dx^2} = -6+6x$

The maximum value occurs when $\dfrac{d^2V}{dx^2}$ is negative.

At x = 1.577,

$\dfrac{d^2V}{dx^2} = -6+6(1.577) = 3.462$

This is positive, so it is minimum.

At x = 0.423,

$\dfrac{d^2V}{dx^2} = -6+6(0.423) = -3.462$

This gives a negative value. Therefore, it is a maximum.

Hence, the maximum volume is obtained when x = 0.423 m.