You need to construct fence around an area of 3136 ft2. What are the dimensions of the rectangular pen that minimize the amount
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<h3>Answer: 4w^2+190w</h3>
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Explanation:
Check out the attached image below. Specifically, take a look at figure 1. We start with a rectangle which is 70 meters by 25 meters. Then we extend this rectangle out by adding two copies of w to each dimension having the larger rectangle be (70+2w) meters by (25+2w) meters.
The original smaller rectangle has area of 70*25 = 1750 square meters.
The new larger rectangle area is a bit more tricky to figure out, though its not too bad once you get the hang of it. You could use the FOIL rule to get
(70+2w)(25+2w) = 70*25+70*2w+2w*25+2w*2w
(70+2w)(25+2w) = 1750+140w+50w+4w^2
(70+2w)(25+2w) = 1750+190w+4w^2
(70+2w)(25+2w) = 4w^2+190w+1750
Or you could use the box method shown in figure 2.
The idea is to write the terms of (70+2w) and (25+2w) along the edges of the box. Then filling out the 4 inner boxes involves multiplying the outer terms. Example: row1,column2 has 50w inside it because we multiply the outer terms 25 and 2w to get 25*2w = 50w. The other boxes are filled out in a similar fashion.
Although the FOIL rule is handy and taught in a lot of math books/classes, I prefer the box method because the box method can be extended to other cases (not just multiplying two binomials).
Whichever method you use, the larger rectangle has area 4w^2+190w+1750 square meters.
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So,
A = area of larger rectangle = 4w^2+190w+1750
B = area of smaller rectangle = 1750
C = difference in areas of A and B
C = A - B
C = ( A ) - ( B )
C = ( 4w^2+190w+1750 ) - ( 1750 )
C = 4w^2+190w+1750-1750
C = 4w^2+190w
This represents the area of just the tile border alone. We started with the larger rectangle and took out the smaller rectangle inside to only have the border left over.
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If you want, you can factor out a 'w' to get
4w^2+190w = w(4w+190)
Though this may be optional.