* Any insight would be greatly appreciated! This is due tomorrow, so.... please

Question

* Any insight would be greatly appreciated!  This is due tomorrow, so.... please help!

A flat, rectangular piece of cardboard is 18” by 12”. A box is formed by cutting squares from each corner of the same size and folding up the sides. Using the side-length of cut-out squares as the independent variable, find an expression for the resulting volume of the box, give a graph over the full domain, and use the graph to estimate the side-length that gives maximum volume, accurate to the nearest quarter-inch.

Explanation / Answer

ok buddy so here is the thinking. you are going to start with a sheet 12 by 18 first you want to to find an equation, so thinking about cutting a distance x from each corner. this will make it so that you have cut out 4 equal squares all with side length x. now what does that mean for your volume. The volume of a box is base * height* width. we call the 18 our width and 12 our base. so if you cut these squares at length x from each corner that means we will have new side lengths( no longer 18 and 12) so our new side lengths are (18-2x) and (12-2x) it is 2 x because we removed squares with side length x from both sides. and if we were to fold these corners up we will have a height of x.

so our final equation for volume is (18-2x)(12-2x)(x)

if you have taken calculus you can multiply this out and take a derivative to find the max volume, or if you want to just use a graphing calculator and find the max. the domain is x can not be greater than 6 or less than 0 because you would then have less than 0 for your side length that started at 12.

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