A rectangular box which is open at the top can be made from an12-by-18-inch piece of metal by cutting a square from each corner and bending up the sides. Find the dimensions of the box with greatest volume, where h = height, l = length, and w = width. (Note: let the width be determined by the12-inch side and the length by the18-inch side.)

Answer:h=2.35in, l=13.3in and w=7.3inStep-by-step explanation:In order to solve this problem, we must start by drawing a diagram that will represent the prolem. (Look at attached picture)From the diagram we can see that the length ad the width of the box are found by  using the following expressions:L=18-2xW= 12-2xH=xnext, we can make use of the volume formula of a rectangular prism, which is the following:V=WLHso we can substitute the expressions we got before on the formula:V=x(12-2x)(18-2x)and for ease of calculation, we can multiply the terms, so we get:[tex]V=x(216-24x-36x+4x^{2})[/tex][tex]V=x(4x^{2}-60x+216)[/tex][tex]V=4x^{3}-60x^{2}+216x[/tex]So now we can find the derivative of the volume:[tex]V'=12x^{2}-120x+216[/tex]and set it equal to zero, so we get:[tex]12x^{2}-120x+216=0[/tex]and now we can solve. We can start by factoring the left side of the equation, so we get:[tex]12(x^{2}-10x+18)=0[/tex]which yields:x^{2}-10x+18=0This one can only be solved by using the quadratic formula:[tex]x=\frac{-b\pm \sqrt{-b^{2}-4ac}}{2a}[/tex]so we substitute the corresponding values:[tex]x=\frac{-(-10)\pm \sqrt{-(-10)^{2}-4(1)(18)}}{2(1)}[/tex]which yields:x=7.65in or x=2.35inwe keep the 2.35in only, because the 7.65in returns an impossible box, since that would make the width a negative width.So the 2.35in height would give us the greatest box. Now we can use it to find the height, the width and the length of the box.Height=x=2.35W=12-2x=12-2*2.35=7.3inL=18-2x=18-2(2.35)=13.3in