Expert: Alon Mandes - 5/12/2009

Question
A rancher wants to fence in an area of 1000000 square feet in a rectangular field and then divide it in half with a fence down the middle parallel to one side. What is the shortest length of fence that the rancher can use?

We only covered word problems for about 15 min in class, so I'm all confused. Thanks for helping!


also, this problem is confusing also..


The top and bottom margins of a poster are 4 cm and the side margins are each 2 cm. If the area of printed material on the poster is fixed at 388 square centimeters, find the dimensions of the poster with the smallest area.

width?
height??

Answer
Suppose that x is the length of the field & y is the width .
Therefore, the amount of fence will be : 2(x+y)+y=2x+3y .
We know that xy=1000000 , thus y=1000000/x. Now our amount will be
A(x)=2x+3000000/x . To find minimum we derive A & set it equal zero:
A'(x)=2-3000000/x²
A'(x)=0 -> 2-3000000/x²=0 -> 2=3000000/x² -> x=1224.74
Let's find y : y=1000000/x=y=1000000/1224.74=816.5
Hence A(minimum)=2x+3y=2(1224.74)+3(816.5)= 4898.98
-----------------------
The area of printed material on the poster is fixed at 388 , that
means (x-4)(y-2)=388 -> y=388/(x-4)+2 .
The area of the poster is A=xy=2x+776/(x-4). To find the smallest
we need to derive A & set the derivative equals zero :
A'(x)=2-766/(x-4)²
A'(x)=0 -> 2=766/(x-4)² -> (x-4)²=388 -> x-4=19.7 -> x=23.7
Let's find y : y=388/(x-4)+2= 388/(23.7-4)+2 = 21.7

Alon.