Question

from a square piece of metal, whose side is 27 cm long, an open box with square base is to be made. What dimensions should the box have in order to contain the largest volume?

Answer 1

Volume is the (Length)x(width)x(height). Assuming that you are cutting a square out of each corner to make the tabs, label the distance of the cut as X. Then make an equation, the Base of the box has length of (27-2x) and the width of the base is the same (due to the fact that it's square). So the base area is (27-2x)^2. then multiply by the height of the folded box, which in this case should be the X that you cut in for each tab. Therefore the volume should be (X)(27-2X)(27-2X) simplify that out as (X) (729-108X+4X^2) and finally to the equation V=4X^3-108X^2+729X. to make sure you find the maximum of the volume you need to find the derivative with respect to X (dV/dX) and set that equal to 0. You need that zero to be when the derivative goes from a positive to a negative value, Meaning that the rate of growth is increasing, reaches it's max when dV/dX is zero, and then begins to decrease; as you should expect. The derivative is 12X^2-216X+729, Solving for the Zero you use the quadratic equation, which is a mess, or just a graphing calculator. The X value that has the zero rate of change is at X=4.5". Note: the derivative is quadratic, and will cross the x axis twice, you need the first x value of 4.5 not the second value of 13.5, if you think about it you can't cut a square of 13.5 inches out of each corner of a 27 inch sheet, because there would be no material left to make the box.

Hope that helps.