Answer to Question #149215 in Calculus for Kate

Question

Bentwood boxes are made out of one piece of wood that is steamed and bent to form a box (tops and bottom of the box are made separately). 

Suppose you have a cedar plank of length 36 inches and width 3 inches. For the purpose of this problem, suppose the thickness of wood is negligible.
* (a) If you use this plank for the sides of the box only, find the dimensions of the largest (in terms of volume) rectangular bentwood box that you can make. 

* (b) Would your answer to part a) change if the width of the plank was different? Explain why or why not. 

* (c) Suppose now that you want to use this plank to make the sides and the bottom of the box. You are only allowed to make one cut to cut off a piece of the plank for the bottom of the box; you will then bend the rest of the plank to form the sides of the box. Find the dimensions of the largest (in terms of volume) bentwood box that you can make. 


Answer 1

a) Volume V of the box equal: V=Sh, where S - area of the bottom, h- height of the box. h equals the width of plank, h=3. It is constant. So volume of box is determined by the area of its bottom S.

Let x- length of one side of the bottom. y - other side of the buttom. S=xy

Derivative S_x shows length x, when S is maximum. S_x=y. So S will be maximum, when x=y.

Box perimeter P equals the length of plank. P=36=2x+2y.

x+y=18

x=y=9 inches.

So the largest volume equals V_max= xyh=9x9x3=243 inches³

b) If width of plank is changed, the maximum volume of box will change because it is a multipler in the volume formula. Dimeshions of the buttom (x,y) will not change in this case because it does not depend on the width of plank.

c) If we need to make a bottom of plank, we should divide it by 3 equal long parts and two parts with length equal 3 inches because the shortest side if the bottom will be equal to the width of plank, 3 inches. So 36=3+3+3x

x=10 inches, y= 3 inches, h = 3 inches.

V_max=xyh=10x3x3=90 inches³