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

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,

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

Simplify by dividing by 4.

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

The maximum value occurs when is negative.

At x = 1.577,

This is positive, so it is minimum.

At x = 0.423,

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

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

21

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