Can you help me find a solution

Question

you are designing a box without a too by cutting squares of equal size from the corners of a piece of cardbox measuring 12 centimeters by 8. after moving squares you fold up the sides.

Answer 1

Create a diagram to verify that the 3-d rectangular box created in this way will have dimensions x by (8 - 2x) by (12 - 2x), where x is the side length of the squares cut out from the four corners.

For this reason, the volume is given by V(x) = x · (8 - 2x) · (12 - 2x) = 4x³ - 40x² + 96x ; 0 < x < 4

Often, we now use a graphing utility to determine the dimensions of the square that maximize this volume and the maximum volume that can be attained. These values are the coordinates of the local maximum that exists in the valid domain, 0 < x < 4.

Answer 2

You didn't ask a question. The typical question here (normally, a Calculus question) is to maximize the volume. If you draw a picture, you see that the height becomes x and the base becomes 12 - 2x by 8 - 2x

Thus, the volume is x(12 - 2x)(8 - 2x) = 4x(6-x)(4-x) = 4x(x² - 10x + 24) or 4x³ - 40x² + 96x

Note that 0 <= x <= 4

In Calculus, you take the derivative (slope at a point, if you will) and set it equal to 0.

f'(x) = 12x² - 80x + 96 = 4(3x² - 20x + 24)

If 3x² - 20x + 24 = 0,

x = (20 ± √112)/6 = (10 ± 2√7)/3

The larger value will be a minimum and outside the range [0,4] - you can see this by taking the second derivative or looking at the shape of the cubic.

The lower value is the solution, (10 - 2√7)/3