Expert: Alon Mandes - 4/28/2009

Question
1. A large bin for holding heavy material must be in the shape of a box with an open top and a square base. The base will cost 6 dollars a square foot and the sides will cost 8 dollars a foot. If the volume must be 110 cubic feet. Find the dimensions that will minimize the cost of the box's construction.

Base :
Each side :

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A farmer wishes to build a large rectangular grazing area of dimensions, width, w, and length, l,splitting the area in to 4 equal fields by dividing the width into 4 equal pieces. The total area of all the fields must be 10000 square feet. Find the dimensions that uses the least amount of fencing.

Width w =
Length l =

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A rectangular page is to contain 155 square inches of print. The top and bottom margins are each 0.8 inches wide , and the margins on each side is 1.5 inches wide. What should the dimensions be if the least amount of material is to be used

Length of top(bottom) : ___ inches
Length of side : ___ inches

Answer
I'll solve the 1st problem & I will leave the other 2 , for you as
an exercise :
The Area of the base is x² & the coast is 6x²$ .
The Area of the side  is xy & the coast is 8xy$ .
The total coast is : C($)=6x²+4*(8xy)=6x²+32xy .
The constrain on the volume is x²y=110 -> y=110/x² .
C(x)=6x²+32x(110/x²)=6x²+3520/x .
To find minimization we derive :
C'(x)=12x-3520/x²
Now we solve C'(x)=0
12x-3520/x²=0
12x=3520/x²
x³=293.33
x=6.644
& y=110/44.14=2.49

Alon.