OK!! So with this problem we have to find the maximum volume of a box...but it gets tricky!!
Alright, so you have a piece of cardboard that is 12 by 14cm. You are going to construct an open box (no lid) by cutting congruent squares out of the corners and then folding the sides up.
What are the dimensions of this box and what is the max volume?
Please show work because I really need to understand this!!
thnx!!
2006-12-12 15:37:15 · 2 answers · asked by Seanoso88 1 in Science & Mathematics ➔ Mathematics
You need to write the volume as a function of what you cut out, then take the derivative, set = 0 to find the max (of course you should test this critical point to be an actual max)
Let x be the length of the side of the cut out square. The remaining sides of the folded up box are 12-2x and 14-2x (2 squares cut out per side). The volume is the area of the base (12-2x)(14-2x) times the height, which is just x.
Expand the polynomial to simplify taking the derivative, which will be a quadratic that you can easily solve.
2006-12-12 15:45:13 · answer #1 · answered by grand_nanny 5 · 0⤊ 0⤋
The volume is:V(x) = (12-2x)*(14-2x)*x
Find the maximum by differentiation and equating to zero:
dV/dx = 4(42 - 26 x +3x^2) = 0 => x1,2 = (1/3) (13 -/+ Sqrt[43])
x2 is greater than one of the sides therefore only x1 is a possible solution: x1 = 2.1475.
The volume of the box is 160.584 cm.
2006-12-12 23:58:41 · answer #2 · answered by Boehme, J 2 · 0⤊ 0⤋