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January 4th, 2010, 04:24 PM   #1
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HELP w/ App. of Derivative Problem???

A design team wants to determine a design of a cereal box that MINIMIZES THE TOTAL COST OF MAKING ONE. The girth of each box must be no larger than 20in., and the height is fixed at 10in. Because cardboard costs money, you have the following unit costs for materials:

Faces: $.017 per square inch
Sides: $.04 per sq. inch
Top & Bottom: $.01 per square inch

Write an expression for the cost of making a cereal box, then find the dimensions of the cereal box that will MINIMIZE THE COST.

***

any or all help appreciated...thanks SO MUCH <33
i figured out the following:
Look at the 3 dimensions of the box we know 2L+2W =20
or L+W = 10
so if L=x ,W=10-x

Height = 10
Length = x
Width = 10-x

Areas & costs
Faces = 2*0.017* (x)(10)
Sides = 2*0.04* (10-x)(10)
Top&B = 2*0.01* (x)(10-x)

Total cost
C=2*0.017* (x)(10)+2*0.04* (10-x)(10)+2*0.01* (x)(10-x)

[color=#FF0000]BUT I KEEP GETTING A NEGATIVE ANSWER FOR MY "X" CRITICAL POINT...AND WHEN I TEST IT, IT'S A MAX INSTEAD OF A MIN. PLEASE HELP FOR WHAT TO DO AFTER I DERIVE THE EXPRESSION.....thanks[/color]
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January 5th, 2010, 05:16 AM   #2
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Re: HELP w/ App. of Derivative Problem???

If you simplify the expression you get
f(x) = - 0.02x - 0.26x + 8
which is a parabola with a maximum when x = -6.5.

However, physical constraints mean that 0 < x < 10.
The function f(x) is decreasing throughout that interval.
So the minimum is f(10).

The result is a useless flat box. This could have been predicted, given that the sides are more than twice as expensive as the faces. (It is possible that a mistake has been made in setting the question, or that there is a misprint.)
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January 5th, 2010, 06:56 PM   #3
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Re: HELP w/ App. of Derivative Problem???

yes, i got x=-6.5 too, which is why i'm confused. thanks for the help...

so, considering that there is no error in the question, was my initial approach at least correct? is x=10 really the only possible solution?? i typed it out exactly as my teacher gave it to us...is it a trick question or something??
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January 5th, 2010, 09:30 PM   #4
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Re: HELP w/ App. of Derivative Problem???

I suppose you could glue together the margins of two pieces of cardboard and stuff cereal into the gap.

If it has to be a cuboid, then the cheapest box would have zero volume. If you had been given a minimum volume V, you could then solve 10x(10-x)=V for x (getting two solutions) and then choose the higher value of x.
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