SOLUTION: a rectangle has a perimeter of 100 cm what is the length and width if the rectangle is to have the largest possible area

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Question 505917: a rectangle has a perimeter of 100 cm what is the length and width if the rectangle is to have the largest possible area
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
The largest area for a given perimeter is always a square,
but you can prove it
let the vertical side = y
Let the horizontal side = x
If the perimeter is P,
and the area is A
+P+=+2x+%2B+2y+
+P+=+100+ cm
+100+=+2x+%2B+2y+
(1) +x+%2B+y+=+50+
+A+=+x%2Ay+
+y+=+50+-+x+
+A+=+x%2A%2850+-+x%29+
+A+=+-x%5E2+%2B+50x+
Since the coefficient of the squared term is negative
this curve has a maximum, not a minimum.
The vertex is at +x+=+-b%2F%282a%29+
+b+=+50+
+a+=+-1+
+x+=+-50%2F%282%2A%28-1%29%29+
+x+=+25+
and, since
+y+=+50+-+x+
+y+=+25+
So the sides are all 25 cm and the max area is a square