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 January 2nd, 2010, 07:23 AM #1 Newbie   Joined: Jun 2008 Posts: 15 Thanks: 0 HELP w/ App. of Derivative Problem??? When cereal is shipped to the supermarket, it usually comes inside a bigger shipping box. Assume the total volume of the shipping box to be 25,000 cubic inches. Given: - the girth of each cereal box is mandated to be no larger than 20in. - height of each cereal box is fixed at 10in. - because height of cereal box will be 10in, 2 stacks of cereal boxes will fit in a shipping box - the shipping team wants to continue to fit 5 cereal boxes on the width - the shipping team wants to MINIMIZE the number of cereal boxes on the length - let "n" be the number of cereal boxes that can fit into a shipping box on the length How do I determine the WIDTH of a cereal box that MINIMIZES the number of cereal boxes that can be fit into a shipping box on the length? *** ANY OR ALL HELP APPRECIATED...thank you SO SO much... <33
January 2nd, 2010, 04:17 PM   #2
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Re: HELP w/ App. of Derivative Problem???

Heloo, dagitt!

Quote:
 When cereal is shipped to the supermarket, it usually comes inside a bigger shipping box. Assume the total volume of the shipping box to be 25,000 cubic inches. Given: - - The girth of each cereal box is mandated to be no larger than 20in. - - The height of each cereal box is fixed at 10in. - - Because height of cereal box will be 10in, 2 stacks of cereal boxes will fit in a shipping box - - The shipping team wants to continue to fit 5 cereal boxes on the width - - The shipping team wants to MINIMIZE the number of cereal boxes on the length - - Let "n" be the number of cereal boxes that can fit into a shipping box on the length How do I determine the WIDTH of a cereal box that MINIMIZES the number of cereal boxes that can be fit into a shipping box on the length?

Each cereal box has length $x$, width $y$, and height $10.$
Code:
       *- - - - - -*
/           /|
/           / |
* - - - - - *  |10
|           |  |
|           |  |
|           |  *
|           | /
|           |/ y
* - - - - - *
x

$\text{The volume of each cereal box is: }\:10xy\text{ in}^3$

$\text{The shipping box (="Case=") will have: }\:\text{length}\,=\,nx,\; \text{width}\,=\,5y,\;\text{height}\,=\,20$

$\text{The volume of the Case is: }\nx)(5y)(20) \:=\:100nxy \:=\:25,000 \qquad\qquad\Rightarrow\qquad\qquad n \:=\:\frac{250}{xy}" />[color=beige] .[/color][color=blue][1][/color]

$\text{The girth is at most 20 inches: }\:2x\,+\,2y \:\leq\:20 \qquad\qquad\Rightarrow\qquad\qquad x\,+\,y\:\leq\:10$

$\text{To have a }minimum\text{ number of boxes, we use }maximum\text{ girth:}$
[color=beige]. . [/color]$x\,+\,y\:=\:10\qquad\qquad\Rightarrow\qquad\qquad y \:=\:10\,-\,x$[color=beige] .[/color][color=blue][2][/color]

Substitute [color=blue][2][/color] into [color=blue][1][/color]:

[color=beige]. . [/color]$n \;=\;\frac{250}{x(10\,-\,x)}$

And that is the function we must minimize . . .Go for it!

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