To begin with we know that the area of a rectangle (which is how this project starts out) is the length times the width. The volume of our final product (a rectangular prism) will be width(though not the same width) times height times depth.
Because a portion of our original width becomes the side and a portion of our original length becomes the new bottom we need to do some figuring to arrive at a formula that will accurately represent this new shape regardless of the size of triangle we cut out to give it depth.
I used 13 and 19 for the width and length in this tutorial but the actual sizes of you project may vary slightly and the math would need to be re-figured from this point.
So the formula for the volume of the purse made by a 13x19 inch placemat is
In order to find the maximum possible volume we take the Volume formula and find the derivative of it.
If we set the derivative equal to zero and solve for X using the Quadratic Equation we will find the value of X which gives us the maximum volume.
Because there are two possible values of X which make the equation zero we next check each value by plugging it into the original Volume equation.
This makes sense referring back to this picture because the depth is 2X.
If X were 8.14 then the depth would be greater than the available width which is impossible. So the only real answer is X= 2.53 which means that the total depth of our project, in order to obtain the maximum volume is 5.06 inches.
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