A square piece of cardboard is 30 by 30 cm. squares are cut out of the corners then folded up to get a box. what dimensions give the largest volume? what is the largest possible volume?
I know it is a cube, with 5 sides, and can get the answer my way but can't seem to using calculus/derivitives like I am supposed to. please help
2007-12-03 12:53:18 · 2 answers · asked by The Anonymous 1 in Science & Mathematics ➔ Mathematics
All right, I'll help you. If you cut out 4 squares of side x from each corner, the remaining uncut edges would have a length 30 - 2x. The volume of the box formed by folding up the flaps would be the square of this length, times the height, which will be x. So, we have:
V = (30 - 2x)² x
Expanding this, and differentiating this with respect to x, we have
dV/dx = 900 - 240x + 12x² = 0
Divide both sides by x to reduce it to a quadratic equation, which you can solve to find that x = 5 cm. Hence, the dimensions of the box formed would be 20 x 20 x 5 = 2000 cu cm. Notice that it's not a cube, which would have dimensions 10 x 10 x 10 = 1000 cu cm, a smaller volume.
2007-12-03 13:23:01 · answer #1 · answered by Scythian1950 7 · 0⤊ 0⤋
Optimization Box Problem
2016-11-04 10:48:36 · answer #2 · answered by manciel 4 · 0⤊ 0⤋